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ABSTRACT
Title of Thesis: Forecasting Hotel Occupancy Rates in EgyptUsing Time Series Models
Degree Candidate: Mohamcd A. Ibrahim
Degree and Year: Master of Science. 1995
Thesis directed by: Dr. Emmanuel T. AcquahAssociate Professor. Department of Agriculture,University of Maryland Eastern Shore
Key words: Forecasting. Time Series, Occupancy Rate, Market Segments,Hospitality. Accuracy
Forecasting occupancy rates is important for effective hotel management. This
study examines empirically the use of five time series models (moving average. Brown’s.
Holt’s, Winter’s and AR1MA) in forecasting hotel occupancy rates in Egypt. These
models were fitted and tested using actual monthly occupancy rates for two types of
hotels (city center and resort). For the city center hotel. ARIMA model forecasted more
accurately than the other investigated models. Results were different for the resort hotel
where Brown’s model outperformed all ether models in prediction.
Results of this study showed generally that simple time scries models (Brown’s,
Holt’s & moving average) performed as well as or better than sophisticated models
(Winter’s & ARIMA) in forecasting hotel occupancy rates in Egypt. Since simple time
scries models arc easy to develop and economical in terms of time and skill levels of
users, results of this study have important implications for the use of time series
forecasting techniques in the hotel industry.
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FORECASTING HOTEL OCCUPANCY RATES IN EGYPT USING TIME SERIES MODELS
by
MOHAMED A. IBRAHIM
Thesis Submitted to the Graduate Faculty of the Department of Agriculture at the University
of Maryland Eastern Shore in partial fulfillment of the requirement
for the degree MASTER OF SCIENCE
1995
Advisory Committee:Emmanuel Acquah, Ph.D, Advisor Imtiaz Ahmad, Ph.D John Dienhart. Ph.D Frank Lin, Ph.D
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DedicatedTo my father Professor Abdalla Fathy Ibrahim
who did not live to read this work.
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ACKNOWLEDGEMENT
I would like to start by thanking Allah for giving me the knowledge, effort and
patience to complete this study. I would also like to express my gratitude to Dr.
Emmanuel Acquah my advisor and the other members of graduate committee, Drs.
Imtiaz Ahmad, John Dienhart and Frank Lin for their support and advice during the
preparation of this manuscript.
My deep appreciations are also extended to Drs. William Hytche, Mortimer
Neufville, Yousef Hafez and George Shorter for all their support during my study at the
University of Maryland Eastern Shore.
Special acknowledgement to my mother Nagat and my family Omneya, Dalia,
Sarah and Hakam for their prayers, love and support during the preparation of this work.
My gratitudes are extended to my colleagues Greg Early and Richard Coe for
their friendship and help in preparing this manuscript.
Mohamed A. Ibrahim
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TABLE OF CONTENTSPage
List of Tables viList of Figures vii
CHAPTER 1. Introduction 1Statement of the Problem 3Objectives 3
CHAPTER 2. Review of Literature 4Judgmental forecasting 4Econometric forecasting 5Time series forecasting 9
CHAPTER 3. Methodology 17Development of Questionnaire 17Sample 17Data Analyses 19
Measuring the Accuracy of Forecasting Models 19Time Series Models Formulation 20Moving Average Model 20Exponential Smoothing Models 21Autoregressive Integrated Moving Average Models 28
CHAPTER 4. Results and Discussion 36Time Scries Models Fit 37Time Scries Models Forecasts 40
CHAPTER 5. Conclusions and Recommendations 50
REFERENCES 53
APPENDIXES 57A. Questionnaire 58B. Time series plots of models fitting for Cairo hotel 66C. Time series plots of models fitting for Hurgada hotel 72
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LIST OF TABLES
Tables Page
1. Unconditional Least Squares ARIMA Model for the Cairo Hotel 38
2. Unconditional Least Squares ARIMA Model for the Hurgada Hotel 39
3. Six Months Actual Occupancy versus the Forecast of 41 Time Series Models for the Cairo Hotel
4. Six Months Actual Occupancy versus the Forecast of 42 Time Series Models for the Hurgada Hotel
5. Sum of Squared Errors of Forecast (SEE) for the Cairo Hotel 49
6. Sum of Squared Errors of Forecast (SEE) for the Hurgada Hotel 49
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LIST OF FIGURES
Figures Page
1. Time Series Plot for the Cairo Hotel 26
2. Time Series Plot for the Hurgada Hotel 27
3. Autocorrelation for the Cairo Hotel 32
4. Partial Autocorrelation for the Cairo Hotel 33
5. Autocorrelation for the Hurgada Hotel 34
6. Partial Autocorrelation for the Hurgada Hotel 35
7. The Forecast of the Best Models to Forecast for the Cairo Hotel 44
8. The Forecast of the Best Models to Forecast for the Hurgada Hotel 45
9. SSE of Time Series Model Forecasts for the Cairo Hotel 46
10. SSE of Time Series Model Forecasts for the Hurgada Hotel 47
11. Moving Average Model Fit for the Cairo Hotel 67
12. Brown’s Model Fit for the Cairo Hotel 68
13. Holt’s Model Fit for the Cairo Hotel 69
14. Winter’s Model Fit for the Cairo Hotel 70
15. ARIMA Model Fit for the Cairo Hotel 71
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LIST OF FIGURES (continued)
Figures Page
16. Moving Average Model Fit for the Hurgada Hotel 73
17. Brown’s Model Fit for the Hurgada Hotel 74
18. Holt’s Model Fit for the Hurgada Hotel 75
19. Winter’s Model Fit for the Hurgada Hotel 76
20. ARIMA Model Fit for the Hurgada Hotel 77
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CHAPTER 1
INTRODUCTION
Hold management companies that want to meet future challenges of the
hospitality industry must take positive actions toward adopting accurate forecasting
methods. Accurate forecasting of hotd occupancy rates is very important in all areas of
hotd operations. Short term occupancy rates are needed for daily and weekly employees
scheduling; especially where hotds are highly dependent on part time labor. To
maximize revenues, hotd managers must adopt yidd management strategies which
depend on accurate forecasting of slow and peak occupancy seasons and plan to operate
on maximum capacity in all seasons. Marketing strategies, menu devdopment decisions,
employee hiring, training programs and capital investment decisions must be based on
predicting future hotel occupancies accurately.
To address the critical need for forecasting in the hospitality industry, the majority
of hotd operators devdop their own forecasting methods depending on simple averages
of historical business observations, judgements based on experience, or sometimes a mix
of both methods. Hospitality managers use their own judgment based on their experience
of the economic and business activity in estimating future sales (Crange <fc Andrew
1992). The studies that have compared judgmental with quantitative forecasting indicated
that judgmental forecasting is less accurate than quantitative forecasting. Geurts and Kelly
(1986) conducted a study to forecast retail sales, they concluded that judgmental
forecasting is inferior to time series or econometric short term sales forecasts.
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Makridakis (1986) indicated that people’s judgement is one of the obstacles for
successful implementation of forecasting. He reported that individuals prefer making
forecasts judgementally. People believe that their knowledge of the product, market and
customers as well as their insight and inside information give them a unique ability to
forecast judgementally. The author indicated that individuals do not usually consider that
quantitative methods can do an adequate job. He concluded in his study that when a
number of frequently required forecasts are needed, the judgmental method is not a good
practice due to its inaccuracy compared with quantitative methods.
Andrew, Crange and Lee (1990) reported that the application of quantitative
methods in forecasting has been ignored by the hotels managers and educators. This may
be due to the lack of experience in the use of quantitative methods in forecasting or the
lack of evidence of its accuracy in hotels applications.
The cost of judgmental forecasting errors in hotels can be high due to many
reasons such as the loss of food and beverage because of over production, high inventory
costs due to over stocking, loss of customers to competition due to over estimation of
prices in slow occupancy periods, and high labor costs due to over staffing and
scheduling of employees.
Quantitative methods in forecasting takes management from the uncertain
environment into the probability of what might happen in the future by looking at
different variables that affect a dependent variable, or by looking at the historical patterns
and trends over a period of time in the dependent variable in order to capture these
patterns and trends to project the future.
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Problem Statement
The problem is the inability of hotels to forecast occupancy rates accurately.
Purpose
This study will provide evidence of the efficiency and accuracy of quantitative
methods, namely time series models, in forecasting hotels occupancy rates in Egypt.
Objectives
The objectives of this study were to:
(i) Forecast hotel occupancy rates in Egypt using moving average, exponential
smoothing, and autoregressive integrated moving average (ARIMA) time series
forecasting models.
(ii) Determine which of the time series models is more accurate in forecasting hotel
occupancy rates in selected hotels.
(iii) Determine the effect of type of hotel operation, location and market segmentation
on forecasting errors of the models.
(iv) Determine which of the time series models would be most appropriate for use by
hospitality managers based on accuracy and simplicity.
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CHAPTER 2
LITERATURE REVIEW
Judgmental Forecasting
Crange and Andrew (1992) reported that the judgmental forecasting is the
technique most often used by the majority of hotels and restaurants. The judgmental
technique consists of an intuitive forecast based on the manager's collective experience
regarding the variable in question (for example occupancy rate).
Makridakis and Wheelwright (1989) reported the disadvantages of judgmental
forecasting. The authors reported that humans tend to be optimistic and underestimate
future uncertainty. They indicated that the reasons for the poor performance of
judgmental forecasters are because humans are not capable of separating wishful
thinking, politics, personal consideration, and their own emotional state from an objective
evaluation of the past, present or future.
Simon and Newell (1971) discussed the reasons for the poor performance of
judgmental forecasting methods. They pointed out that because people have limited
information processing capacity, they can not deal directly with large problems. Instead
they use a simplifying heuristic approach in their problem solving efforts. In the end,
they look for satisfying, rather than optimizing, solutions since in the complex
environment in which we live, optimizing is well beyond present human intellectual
abilities.
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Mabert (1975) reported a direct comparison between judgmental and quantitative
methods of forecasting. He found that forecasts based on opinions of the sales force and
corporate executives in different industries gave less accurate results over the five year
period covered by the study than did the quantitative forecasts of three methods
(exponential smoothing, harmonic smoothing and ARIMA). In a comparison between the
performance of judgmental and quantitative methods in forecasting, Adam and Ebert
(1976) found that exponential smoothing method produced forecasts that were statistically
more accurate than those of human forecasters.
Many studies have been conducted on the accuracy of judgmental versus statistical
models. For example, Armstomg (1983), Lorek and Patz (1976), Cleveland and Tiao
(1976), Hogarth (1975) and Dalrymple (1975), all concluded that quantitative methods
provided better forecasts than judgmental methods.
Econometric Forecasting
In econometric forecasting models, the value of a certain dependent variable is
a function of one or more other independent variables. Makridakis & Wheelwright (1989)
indicated that the strength of an econometric model used as a forecasting method is that
a manager can develop a range of forecasts corresponding to a range of values for the
different independent variables. They also reported on three main disadvantages of
econometric models. First, it requires large amounts of data and information about all
the variables used in the model. Information on several independent variables and the
dependent variable must be collected to build the model, the extensive data collection
makes econometric models expensive and time consuming. Secondly, since explanatory
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models generally relate several factors, they usually take a long time to develop and are
more sensitive to changes in the underlying relationships. As a result, they must be
constantly revised and redeveloped according to changes in the relationship. Finally, they
require an estimation of future values of the independent variables before the output can
be forecasted. All these factors make the econometric models time consuming in
collecting the data and difficult to develop, especially the model identification and
estimation which requires a full understanding of economic theory, statistics and
econometrics.
Steckler (1968) conducted a study to predict economic activities namely inflation,
using econometric models. The author reported that the results suggest that econometric
modes have not been entirely successful in forecasting economic activity. Similarly,
Cooper and Jorgenson (1969) conducted a study to predict the components of the national
income and product accounts by using econometric and exponential smooching models.
They indicated that the econometric models are generally not superior to purely
smooching methods of forecasting.
Naylor and Seaks (1972) conducted a study to forecast investment using
econometric and ARIMA models. The authors reported that few forecasting quarterly
investment from 1963 to 1967 the ARIMA forecasts were superior to econometric models
in forecasting quarterly investment rates based on average absolute error.
Cleary and Fryk (1974) conducted a study to forecast the telephone demand using
econometric model versus time series models namely ARIMA. They concluded that
ARIMA models outperformed the econometric model in forecasting telephone demand.
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Binroth, Burshtein, Haboush and Hartz (1979) conducted a study to forecast
monthly rubber commodity prices using time series versus econometric models. They
concluded that the ARIMA time series model used was superior to the econometric model
in forecasting monthly rubber commodity prices. The authors reported that the advantage
of ARIMA model over the econometric model comes from the ability of ARIMA models
to elucidate the various components of time series such as non-stationarities (trends) and
cyclic, seasonal and random effects better than econometric models. They indicated the
disadvantages of econometric models are because they require the user to determine
which predictor variables to use, which transformations of the input data to perform, and
also require the forecasts of predictor variables that are not leading indicators of the
independent variable. The human estimation of the model could lead to identification
errors which increase the forecast errors. The automatic methods namely time series
eliminate human intervention in the model identification and estimation which eliminate
human errors and lead to accurate forecasts.
Schmidt (1979) conducted a study to compare the performance of econometric
versus time series models, namely ARIMA, in forecasting monthly retail sales. The
author concluded that the ARIMA model performed better than the econometric model
in forecasting monthly retail sales.
Geurts and Kelly (1986) conducted a study to forecast sales in retail stores. The
authors used two time series models versus an econometric model to forecast monthly
sales. They reported that there were many disadvantages in using econometric models.
First, the future values of the causal variables had to be predicted. Second, in the case
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of using a lagged causal variable to predict sales one period ahead, the reported causal
variable value is often a preliminary figure that is later revised. Both of these factors can
cause data in an econometric model to be inaccurate and the model to be weak in its
ability to forecast. Third, the continual need to gather data can make these models time
consuming and expensive to use. And fourth, causal relationships can change over time,
making it necessary to constantly update or totally redesign the model. They concluded
in their study that time series models namely exponential smoothing and ARIMA are
superior to econometric models in retail sales forecasting.
Crange & Andrew (1992) conducted a study to forecast restaurant monthly sales
using econometric versus time series models. They reported that econometric models
utilize a regression equation or equations to establish a causal relationship between the
dependent variable (hotel occupancy) and independent variable such as disposable
income, consumer price index and unemployment. They indicated that the advantage of
econometric models over time series models is that the econometric model may respond
more quickly in its predictions to the changed situation than the time series models,
especially when there is a turning point in the economy due to unexpected economic
events. They concluded in their study that the time series models outperformed the
econometric models in forecasting monthly restaurant sales. The findings are significant
in a fact that restaurant sales are similar to hotel sales (occupancy rates), both are
influenced by the same economic variables and business activities.
The majority of research that compared the performance of econometric with time
series models in forecasting has shown that time series forecasts were superior to
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econometric models in forecasting different dependent variables in different industries.
Other studies that compared the accuracy of econometric models with time series models
Groff (1973). Christ (1975). Narashimhan (1975), Nelson (1972). Mahmoud (1984) and
Makridakis (1986) all concluded that time series methods are generally superior to
econometric models in short term sales forecasting. The significance of the previously
investigated research is that in all the studies conducted in forecasting sales in different
industries concluded that time series models outperformed the econometric models in
sales forecasting.
Time Series Forecasting
The time series model looks for time patterns (trends, cycles, and seasonal
influences) in a single series of data and captures them in mathematical equations. These
mathematical relationships are then used to project into the future the historical time
patterns in the data. Time series models assumes explicitly that the underlying pattern can
be identified solely on the basis of historical data from that series (Crange Sc Andrew
1992). Three of time series models techniques will be used in this study; moving
average, exponential smoothing and ARIMA.
Moving Average Model
The moving average model uses a process of averaging different time series
observations to smooth (average) the random fluctuations in different points of historical
data. In the averaging process the model gives equal weights to all observations in the
time series data. This moving average model is obtained by finding the mean of the most
recent observations of the time series data and using it to forecast the next period. The
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future forecast is simply the average of the most recent observations.
Exponential Smoothing Models
In the exponential smoothing method, the historical data (monthly occupancy
rates), are used to obtain a smoothed value for the series. That smoothed value is then
extrapolated to become the forecast for the future value of the series. Exponential
smoothing operates in a manner similar to that of moving averages by smoothing
historical observations to eliminate randomness.
Makridakis & Wheelwright (1989) reported that the exponential smoothing models
apply an unequal set of weights to past data. These weights decay in an exponential
manner from the most recent data value to the most distant value. The goal of
exponential smoothing is to distinguish between the random fluctuations and the basic
underlying pattern by "smoothing* the historical values. The estimated weights are used
to eliminate the randomness found in the historical sequence and basing a forecast on the
smoothed pattern of the data. One of the advantages of the exponential smoothing over
the moving average is that it gives more importance to recent observations by assigning
high value weights to them than those of earlier observations. This technique makes the
exponential smoothing models require less amount of historical data to forecast the next
data point than the moving average because it assumes that the next point of the series
is a function of the last point multiplied by a certain weight, in addition to a residual. In
forecasting hotel occupancy rates this will be an advantage because the most recent
occupancy rates could be used to prepare the future forecast without the need to have all
the historical time series data.
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Crange & Andrew (1992) conducted a study to forecast restaurant sales using time
series and econometric models. They reported that the exponential smoothing method has
many advantages over the econometric models. First, the data required to build the
model is less than that of the econometric models. Time series models requires only one
variable consisting of historical data observations over a period of time such as, historical
occupancy rates. This makes time series models less expensive and less time consuming
to develop. Secondly, time series models are easy to develop. They require minimum
training in the area of quantitative methods to prepare accurate forecast. Third, the
forecasting method could be constantly repeated by minimum data entry of the latest
historical observation to obtain a new forecast without the need to revise the model as
in the case of causal econometric models if the relationship between variables changes,
the model will need to be revised and extra data collection might be needed. Last,
Andrew et al. (1990), Crange et. al (1992) and Geurts et. al (1986) support the
superiority of the time series models (Exponential smoothing and ARIMA) to other
forecasting models, namely, econometric models in forecasting sales in retail business
and the hospitality industry.
Many studies have been conducted on comparing the performance of three
different forecasting models such as moving averages, exponential smoothing and
econometric. Kirby (1966) found that in terms of month to month forecasting accuracy,
the exponential smoothing methods did the best forecast. In a study conducted by Levine
(1967), the same three forecasting methods examined by Kirby were compared. Levine
concluded that although the moving average method had the advantage of simplicity,
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exponential smoothing offered the best accuracy for short-term forecasting. Other studies
reported by Gross and Ray (1965), Raine (1971) and Krampf (1972) have arrived at
similar conclusions.
Autoregressive Integrated Moving Average
The Autoregressive Integrated Moving Average (ARIMA) models, are a
specialized class of linear filtering techniques that completely ignore causal variables in
making forecasts. The ARIMA is a highly refined curve fitting device that uses current
and past values of the dependent variable to produce accurate forecasts. The ARIMA
methodology is appropriate if the observations of time series are statistically dependent
on or related to each other (Makridakis & Wheelwright 1989).
Makridakis & Wheelwright (1989) reported, the ARIMA method of forecasting
is different from most methods because it does not assume any partial pattern in the
historical data of the series being forecasted. The ARIMA uses an iterative approach of
identifying a possibly useful model from a general class of models. The chosen model
is then checked against the historical data to see whether it accurately describes the
series. The model fits well if the residuals between the forecasting model and the
historical data points are small, randomly distributed, and independent. If the specified
model is not satisfactory, the process is repeated by using another model designed to
improve on the original one. This process is repeated until a satisfactory model is found.
There are three types of ARIMA models. The selection of the proper model
depends on the patterns of time series data being used. The first type is the ARIMA
technique for stationary data in which its average value does not change over time is
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considered auto regressive (AR). This is because the AR model depends on capturing the
relationships among different time series points and uses the weights of different
correlations over a period of time to predict the future. The second type is the moving
average (MA) model which could be applied if the data is not stationary. The MA is
used to smooth, the recent observations and predict the future. The third model is the
integration between both AR and MA to form ARIMA model. If the time series data has
a mix of stationary and non stationary data the ARIMA model is used with both
autoregressive and moving average terms.
In a study conducted by Nelson (1972), a comparison was made between ARIMA
models and structural econometric models. It was found that the ARIMA models were
superior to econometric models in fitting the data and also in forecasting. The output of
ARIMA models had less forecasting errors than econometric models. Cooper (1972)
concluded that ARIMA forecasting models can be constructed to predict economic
variables about as well as econometric models. Naylor and Seaks (1972) conducted a
study to forecast investment. They reported that the ARIMA results were significantly
better than econometric models in forecasting the future in all cases under investigation;
ARIMA provided better forecasts by a factor of almost two to one.
Other studies by Christ (1951), Steckler (1968), Cleary and Fryk (1974), Cooper
and Nelson (1975) and McWhorther (1975) concluded that ARIMA were superior to
econometric models in forecasting future business or economic activities. The suggested
reason for the better performance of ARIMA models was the inability of econometric
models to accommodate structural changes in the economy (Christ, 1951; Steckler, 1968;
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Cooper, 1972; Nelson, 1972).
There is considerable disagreement concerning comparisons between ARIMA and
smoothing methods. Newbold and Granger (1974) indicated that the ARIMA forecasts
did better than two fully automatic forecasting models, the Holt, Winters models and one
model based on stepwise regression. A similar conclusion was reached by Reid (1969)
who reported that the ARIMA method was clearly better than exponential smoothing
model.
Groff (1973) conducted a study to forecast retail sales using time series models.
The author, arrived at a rather different conclusion. He reported that the forecasting
errors of the best of the ARIMA models that were tested were either approximately equal
to or greater than the errors of the corresponding exponentially smoothed models for
most series. Similarly, Geurts and Ibrahim (1975) conducted a study to forecast tourism
demand; they found that the exponentially smoothed models patterned on Brown's model
and the ARIMA approach seemed to perform equally well.
Crange and Andrew (1992), conducted a study to investigate the appropriateness
of various classes of forecasting models in forecasting restaurant sales. The authors used
two time series models; exponential smoothing and ARIMA versus one econometric
model. They concluded that the fitted time senes models provided a degree of accuracy
better than the fitted econometric model. In forecasting seven months of the sales ahead,
the time series models generally performed better than the econometric model. The
researchers reported that the ARIMA model was superior to the exponential smoothing
model in fitting and forecasting seven months ahead. They added that the time senes
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models were easy to develop and use, required minimum data collection, and less
experience to build which made it ideal for the use in the hospitality industry for
forecasting.
Andrew, Crange and Lee (1990), conducted a study in which they forecasted hotel
occupancy rates in a full service city center hotel. They used time series analyses,
particularly the ARIMA model and exponential smoothing. They concluded that the
ARIMA model performed better than the exponential smoothing model in fitting and
forecasting future six months occupancy rate for a full service city center hotel. They
reported that the findings of their study could vary for different types of hotel operations,
and they suggested that this should be investigated.
Based on the findings of Andrew, Crange and Lee (1990) and Crange and Andrew
(1992) the accuracy of time series models (exponential smoothing and ARIMA) is
comparable or better than econometric models in forecasting hotel and restaurant monthly
sales. They are also more economical to develop and implement than econometric models
based on the data requirement and skills required to develop. This will make the time
series models more applicable for use in the short term of daily, weekly and monthly
occupancy rate forecasts (which is the main purpose of this study).
The significance of previously investigated studies support that the time series
models (exponential smoothing and ARIMA), are superior to econometric models in sales
forecasting. However, there is a disagreement on the superiority of exponential
smoothing or ARIMA models in forecasting sales.
The literature supports that the time series models are more accurate in
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forecasting monthly variables in different industries like commodity prices, retail stores
sales and hotels occupancy rates. This study will use time series models (Brown’s, Holt,
Winter’s, ARIMA and moving average), because they have proven to be superior to
econometric models in fit and in forecasting of future hotel and restaurant sales based on
the findings and recommendations of Andrew, Crange and Lee (1990) and Crange and
Andrew (1992).
This study will not include econometric models because the review of literature
revealed that the majority of research that compared the performance of econometric with
time series models in forecasting proved that time series forecasts were superior to
econometric models. Steckler (1968), Groff (1973), Cleary and Fryk (1974), Christ
(1975), Cooper, Nelson (1975) Geurts and Kelly (1986) and Crange and Andrew (1992)
all concluded that time series methods are superior to econometric in sales forecasting.
The second reason behind not including the econometric models in the study is that the
main purpose of the study was to provide a simple accurate forecasting method for the
hotel managers to use in all areas of hotel operations. The econometric models need
special experience and training in the area of statistics, econometrics and economic
theory (not acquired by most hotel managers) which makes it beyond the ability of hotel
managers to use as a management tool for daily, weekly and monthly short term
forecasting.
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CHAPTER 3
METHODOLOGY
The purpose of the study was to provide evidence of the efficiency and accuracy
of quantitative methods, specifically time series models, in forecasting hotels occupancy
rates in Egypt. Time series data of historical occupancy rates was collected from two
hotels in Egypt to fit the time series models and forecast six months beyond the fitting
horizon.
Development of Questionnaire
A questionnaire was developed to collect historical time series data of monthly
occupancy rates. The questionnaire included two sections. The first section included
questions to collect information about the hotel such as size, number of rooms, number
of restaurants, management company and forecasting techniques currently applied. The
second section consisted of tables to collect time series data of monthly occupancy rates
starting January 1991 until June 1994. Appendix A presents a sample of the questionnaire
used for data collection.
Sample
The subjects in this study were two hotels in Egypt. The first was the Cairo
Sheraton hotel and the second was the Hurgada Sheraton hotel. They were selected to
accomplish the objectives of the study according to the following criteria:
(1) Same management company to control for management techniques used in
forecasting; and to address the first objective which was to forecast monthly
occupancy rates for both hotels. Also to achieve the second objective which was to
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find which time series model was more accurate in forecasting the occupancy in the
selected hotels.
(2) The two different locations to address the third objective of the study which was to
fmd-out if the forecast errors will be affected by changing the location of the hotel.
(3) Two different type of hotels (city center, resort) to address the third objective which
was to check if the forecasting accuracy of different models will be affected by
changing the type of hotel. This characteristic also was to accomplish objective four
which was to recommend which was the most appropriate model to be used by the
hotel managers in Egypt based on accuracy and simplicity.
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Data Analyses
Measuring the Accuracy of Time Series Forecasting Models
The word accuracy refers to "goodness of fit," which in turn refers to how well
the forecasting model is able to reproduce the data that is already known. In econometric
models, goodness of fit measures predominate. In time series models, it is possible to use
a sub-set of the known data to forecast the rest of the known data, enabling one to study
the accuracy of forecasts more directly (Makridakis and Wheelwright, 1989).
The sum of squared errors (SSE), statistical measure of forecasting accuracy, was
used in this study to determine which model would best fit the data and forecast the
future. The most accurate model would be the one which had the lowest SSE value.
Sum of squared errors
The SSE is found simply by squaring every forecasted error and totaling all of
the squared errors. The following is the equation to obtain SSE
SSE =E [eJ3
e, = forecast error
The SSE test was used to test for goodness of fit during the process of fitting the
time series models to the data. The best model that fitted the data was the one that had
minimum SEE value. Similarly the same test was used to measure the accuracy of
forecast, and the model that forecasted six months ahead with lowest SSE was the most
accurate.
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Alternative Time Series Models Formulation
Makridakis and Wheelwright (1989) explained the model formulation of different time
series models as follows:
Moving Average Model
Ft*i= S, X i - : (1-1)N
Where
F,*, * forecast for time t + 1
S, = Smoothed value at time t
N = The number of observations included in average
This means that once the forecast for the period t (that is F), was obtained could
obtain the forecast for period t+1 by adding X/N and subtracting X ^ N -
The value of equation (1.1) was called the moving average model and could be modified
to obtain
Ft.,= 2 L * 2 U + F. (1.2)N N
Written in this form, each new forecast based on a moving average was an adjustment
of the preceding moving average forecast. It is also easy to see why the smoothing effect
increased as N became larger, a much smaller adjustment was being made between each
forecast.
In this study, a double moving average was used to smooth (average) the random
fluctuation, trend and seasonality in the data. The first 12 observations were added and
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divided by 12 to get the the second year values, Then the twelve obtained observations
of the second year were averaged again to get the next values.
Exponential Smoothing Models
The technique of exponential smoothing could be developed using Equation 1.2
If one only had the most recent observed value and the forecast was made for the same
period, equation 1.2 could be modified so that in place of the observed value in period
t-N one could employ an approximate value. A reasonable test would be to forecast the
value from the preceding period. Thus equation 1.2 could be modified to give
F«*,=& + E. + F, (1.3)N N
This equation can be written as
Ft. l =_LXt + ( l - J J F t (1-3.a)N N
What is presented in Equation 1.3.a is a forecast that weighed the most recent
observation with a weight of value 1 / S and the most recent forecast with a weighted
value of (1JJ.N
If we substitute the symbol alpha in place of 1/N, the following is obtained
F ,^ * o X, + (1 • o) F, (1.4)
This equation is the general form used in computing a forecast by the method of
exponential smoothing. This equation corrects one of the problems associated with
moving average in that extensive historical data is no longer needed to be stored. Rather,
only the most recent observation, the most recent forecast, and a value of a are required
to prepare a new forecast. If Equation 1.4 is expanded by substituting the value for Ft
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which is equal to F,= a X,., + (1 - a) F,., (the value of one iag behind), we get
Ft«.j= a X, + (1 - a) [a X*, + (1 - a) Fwl]= a X, + o ( l - a)X,., + (1 - a)2 F„ (1.5)
However,F,*,= a Xt.2 + (1 - a) F,.j
If this substitution process is carried out even further, we obtain the relationship
F„ , = o Xt + a (l - or) + (1 - a)2 F„2 + (1 • o)J F„2
Brown’s One parameter model
F,«.j = a X, + a(l • oOX,., + (1 * a)2 X,.2 +
(1 - a )J X,, + (1 - a)4 X ^ (1.6)
From this equation it can be seen that Brown's exponential smoothing model gives
decreasing weights to older observed values; that is , since a is a number between 0 and
1; thus (1-a) is also a number between 0 and 1. the weights a , a (l-a ), a (l-a )2, etc. have
exponentially decreasing values. Hence, the name exponential smoothing is used because
it gives less importance for the old data observations to forecast the next data point.
The method of single parameter exponential smoothing is appropriate when the
data series contains a horizontal pattern and does not have a trend. If Brown's model is
used with data series that contain a trend, the forecast will trail behind (lag) that trend.
To solve this problem two parameters model are used to take into account the presence
of a trend.
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Holt’s two parameters model
T,= B(S, - S,.|)+ (l-B)Tt., (1.7)
where
S, = Equivalent of single exponential smoothed value
B = Smoothing coefficient, analogous to a
T, = Smoothed trend in data series
The principle underlying Equation 1.7 is the same as that of single exponential
smoothing represented by Equation 1.4. The most recent trend, (St-Swl) is weighted by
B and the last smoothed trend T,.,, is weighted by (1-B). The sum of these weighted
values is the new smoothed trend value. Holt’s two parameter model uses equation (1.7)
to obtain a smoothed value of the trend and combines this trend with the standard
smoothing equation to obtain
S,= a X, + (1-a) (Sv,+ T wl) (1.8)
The difference between Equation 1.8 and equation 1.7, is the additional term TM
that is added to S,., to adjust the smoothed values for the trend pattern in the data series.
The model will give weights to the data series that has a trend to remove the impact of
that trend. The best value of a and B are found by trying various combinations of values
between 0 and 1 and then selecting the set values that minimizes the SEE. Holt’s model
is superior to Brown's model only if the time series data has a trend. However, if the
data has both trend and seasonal variation Brown’s model will not be able to smooth the
impact of seasonality in the data and this will result in high forecasting errors due to the
seasonality in the data.
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The data was fitted to Holt’s two parameters model in order to get the best
fit.(Recall that the best fitted model was the one that had minimum SSE). The value of
or and 6 coefficient was repetitively changed until minimum SEE was reached.
W inter’s three parameters model
Winter’s three parameter exponential smoothing model had an extra advantage of
being capable of dealing with seasonal data that has a trend. This model was based on
three equations, each equation smoothed one of the three components: randomness, trend,
or seasonality. The following are equations representing Winter's three parameters model
Equation 1.9 which attempts to smooth the randomness in the data. Equation 1.10
attempts to smooth the trend of the time series data and Equation 1.11 attempts to smooth
the seasonal factor in the time series data.
S ,- <O L+ ( l - a X ^ + T ^ ) (1.9)
T,« B(S, - SVI)+ (i-8)TM (1.10)
it* e _&_+ (i-0) (i.ii)s,
whereS ■ smoothed value of deseasonalized seriesT ■ Smoothed value of trendI ■ Smoothed value of seasonal factorL ■ length of seasonality
Winter's model smoothed each component of the time series separately. This
model was ideal when analyzing data that contained trends and seasonality. This model
used three different coefficients to smooth the data that has a trend and seasonality. First,
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it gave exponential values between 0 and 1 to every observation of the data series a to
smooth the randomness of the data, then it used B to remove the trend effect from the
data and 0 to smooth the seasonality. The integrated three equations (1.9,1.10,1.11)
treated the randomness, trend and seasonality in the data to transform the data into a
stationary data in order to give the best fit. The data was fitted to Winter’s three
parameters model in order to get the best fit. The best fit is the one with the minimum
SEE.
Exponential Smoothing Model Identification
To identify the exponential smoothing models, the data was plotted to detect
different patterns (trend, cycles or seasonality). The data of 36 months (January 91 to
December 93) for both subjects were entered in to the computer and plotted to get a time
series plot. Six months of the data (January 94 to June 94) were held in reserve to be
forecasted and to be used as a test of the accuracy of different time series models. Figure
1 presents a time series plot for Cairo hotel and Figure 2 presents time series plot of
Hurgada data.
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CAIR
O (X
CIJI
'ANC
Y UA
TKS
70-
3 0 -
8 15 22 29 36
JANUARY 9 1 TO DECEMBER 9 3
Figure 1. Time Series Plot for the Cairo Hotel.
Figure 1 indicated that the occupancy for the Cairo hotel is characterized with
increasing trend in the first 12 months until the occupancy reached over 90 percent The
second 12 months are characterized with stability of the occupancy between the range of
90 and 70 percent. The third 12 months are characterized with increasing and decreasing
random shocks (random fluctuations). The change of the behavior of the time senes data
in the three years is due to the different types of market segments in the Cairo hotel
operation. As was indicated in the questionnaire, it is a full service city center hotel,
which gives it the advantage of targeting different market segments. The hotel uses
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different marketing techniques to approach different markets, this cause the occupancy
not to vary in different seasons. This is accomplished by replacing one market segment
with another market segment (for example, travel with corporate) to keep the hotel
occupied at a stable rate.
8 0 -
20
29 36228 151JANUARY 9 1 TO DECEMBER 9 3
Figure 2. Time Series Plot for the Hurgada Hotel.
Figure 2 presented the a time series plot for the Hurgada hotel. The occupancy
is characterized with constant peaks and valleys every 12 months in the occupancy rate
except the last 12 months have many random shocks. These peaks and valleys are
existent almost in every year due to changing seasons because as it was reported in the
questionnaire the hotel depended only on the travel market which is fluctuating according
to different seasons.
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Autoregressive Integrated Moving Average Models Formulation
Autoregressive Model
An autoregressive (AR) model takes the form
Y t = 8, Y t _, + Bj Y t2 + . . . + 6P Y tp + e, (1.12)
Where:
YT — dependent variable (future occupancy rate)
Y t i . Y T 2, Y T.p = independent variables that are dependent variables lagged at specific time periods.
8, , 6;, 8p= regression coefficients
eT = residual term that represents random events not explained by the model
Equation 1.12 introduced autoregressive models. In this equation the regression
coefficients were found by using the nonlinear least squares method which used an
iterative solution technique to calculate the parameters rather than using direct
computation. Preliminary estimates were used as starting points: then these estimates
were systematically improved until optimal values were found. The optimal value is the
one that has the best fit. (minimum SSE). Furthermore, the variance for equation 1.12
was calculated in a manner that takes into account the fact that the independent variables
were correlated with each other. Finally, equation 1.12 does not contain a constant term.
This approach was used because the dependent variable values (Ys) are expressed as
deviations from their mean.
Seasonal Autoregressive Model
A seasonal autoregressive (SAR) model takes the form
Yt = 8, YT1J + Bj YT lJ + . . .+ Bp Yt t + Cr (1.12.a)
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The model in Equation (1.12.a) is similar to AR model but is adjusted according
to the season. In this case the data was considered monthly seasonal data.
Unfortunately, not all data series can be handled with autoregressive models. For
this reason the Box-Jenkins (1976) approach also uses the moving average model.
Moving Average Model
A moving average model (MA) takes the form
Yt = Ct - WUT., - - ....... - W ^ (1.13)
Where:
Yt = dependent variable (future occupancy rate)
W „W 2,W, = weights decaying exponentially
Cj = residual term that represents random events not explained by the model
Equation 1.13 is similar to Equation 1.12 except that the dependent variable YT depends
on previous values of the residuals rather than on the variable itself.
Moving average models provide forecasts of YT based on a linear combination of
past errors, whereas autoregressive models express YT as a linear function of some
number of actual past values of YT. It is customary to show the weights with negative
coefficients, even though the weights can be either positive or negative. The sum of W,
+ W2 + .......+ W, does not need to equal 1 and the values of W, are independent
weights estimated by the model not depending on moving average of the previous
observations to obtain a new observation as they are with the moving average
computation.
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Seasonal Moving Average Model
A seasonal moving average model (SMA) lakes the form
Yt= Cj - W UT.l2 - - .......- (1.13.a)
The model in Equation 1.13.a is similar to the AR but adjusted for the monthly seasonal
data.
Autoregressive Integrated Moving Average Model
In addition to AR and MA models, the two can be mixed, providing a third class
of general models called ARIMA. Equation 1.12 and 1.13 were combined to form:
Yt = 6, Yt ., + Bj Yt_2 + . . .+ BP YT.P + tx
- W UT., - W * „ - .......-W „T„. (1.14)
Seasonal Autoregressive Integrated Moving Average Model
In addition to SAR and SMA models, the two can be mixed, providing a third
class of general models called SARIMA. Equation 1.12.a and Equation 1.13.a were
combined again to form:
YT= B, Yj.jj + B} Yt.u + . . .+ 8P YTP +
- W lcT1J - W ^ , , - ........... - W ^ . (1.14.a)
ARIMA (1,1,1,1) means that this model has AR, SAR, MA and SMA terms. In
ocher words the SARIMA model is the seasonal auto regressive integrated moving
average model. Similarly the ARIMA (1,1) model uses combinations of past values and
past errors and offer a potential for fitting models that could not be adequately fitted by
using an AR or an MA model separately.
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Autoregressive Integrated Moving Average Models Identification
Selection of an appropriate model of autoregressive integrated moving average
(ARIMA) could be made by comparing the distributions of autocorrelation coefficients
of the time series being fitted with the theoretical distributions for the various models,
this was done to decide if the model is a moving average, seasonal moving average, auto
regressive, seasonal autoregressive or, ARIMA. Before the selection of the model, the
data was transformed by seasonally differencing to remove the seasonal variation and
non-seasonally differencing to remove the trend.
Autocorrelation
Autocorrelations and partial autocorrelations were used to heip identify an
appropriate ARIMA model for forecasting. They allowed the analyst to identify the
degree of relationship between current values of a variable and earlier values of the same
variable while holding the effects of all other variables (time lags) constant.
The degree of the relationship between variables was measured by the correlation
coefficient which varies between +1 and -1. A value close to +1 implied a strong
positive relationship between the two variables. This meant that when the value of one
variable increased, the value of the other tended to increase also. Similarly, a correlation
coefficient close to • 1 indicated the opposite; an increase in one variable was associated
with a decrease in the other. A coefficient of 0 indicated that the two variables were
unrelated. An auto correlation coefficient is similar to a correlation coefficient except that
it describes the association (mutual relationship) among values of the same variable but
at different time periods (Makridakis Sc Wheelwright 1989).
31
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The data was plotted to get autocorrelation plot. A seasonal fluctuation and trends
were detected in the data which needed a transformation of seasonally and non-seasonally
differrtncing to remove the seasonality and trend. Partial autocorrelation then was
obtained to verify that the data became stationary before fitting it to one of the ARIMA
models.
- 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0LAG
12345678 9
1011
CORK. ♦ -------- «•-------- ♦ -------- ----------0 . S 8 8 >0 . 3 0 5 >0 . 1 3 6 >0 . 0 8 0 >0 . 0 2 5 >0 . 0 7 8 >
- 0 . 0 4 1 >- 0 . 0 9 1 >- 0 . 0 5 1 >- 0 . 1 0 2 >
- 0 . 0 9 5 >
• ft ft ftft**f t *
• ft* ftft
ftftft • ft
• •ft* ftftft
MEAN OF THE S E R I E S 7 3 . 6 5 0 6STD . DEV. OF S E R I E S 1 7 . 8 4 2 6NUMBER OP CAS ES 36
Figure 3. Autocorrelation Plot For The Cairo Hotel.
Figure 3 presented autocorrelation plot of the Cairo hoed. It indicated that the
values of correlations were decreasing exponentially until a point where they reverted.
This indicated that the data fit one of the seasonal moving average modds of ARIMA.
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- 1 . 0 - 0 . 3 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0LAC CORR. --------- -- ------«.-------- ---------- ♦ _ -------------- ------------+ -------- +
1 0 . 5 3 8 >2 - 0 . 0 6 2 > * * * <3 - 0 . 0 2 8 > • * <4 0 . 0 3 9 > • * <5 - 0 . 0 4 1 > • • <6 0 . 1 1 9 > • • • • <7 - 0 . 2 0 1 <8 - 0 . 0 0 3 > • <9 0 . 0 7 4 > * * * <
10 - 0 . 1 6 3 > * <11 0 . 0 6 2 > • • • <
MEAN OF THE S E R I E S 7 3 . 6 5 0 6STD. DEV. OF S E R I E S 1 7 . 8 4 2 6NUMBER OF CASES 36
Figure 4. Partial Autocorrelation for the Cairo Hotel
Figure 4 presented partial autocorrelation plot for the Cairo hotel, it showed the
relations between specific lags holding all others constant; it indicated that after seasonal
and non-seasonal differencing transformation the relationship among data points became
random and valid to be fitted using the ARIMA model.
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- 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0LAG CORR. ———f
1 0 . 3 5 9 > * * • * • • * * > *2 0 . 2 5 2 > * * * * * * * <3 0 . 0 6 5 > * * * <4 - 0 . 0 3 6 > * * <5 - 0 . 3 0 1 > * * * * * * * * * <6 - 0 . 3 5 7 > * * * * * * * * * * <7 - 0 . 1 7 7 > * * * * * <8 - 0 . 2 1 5 > * * * * * * <9 0 . 0 2 5 > * * <
1 0 0 . 0 0 7 > * <1 1 0 . 1 7 3 > * * * * * <
KEAN OF THE S E R I E S 6 2 . 1 3 8 9STD . DEV. OF S E R I E S 1 7 . 4 8 3 6NUMBER OF CASES 3 6
Figure 5. Autocorrelation Plot for the Hurgada Hotel.
Figure 5 presented the autocorrelation plot for the Hurgada hotel. The correlation
for different time lags indicated that the first three observations are related and the
correlation decrease exponentially. Then the relation reverts into a negative correlation
with random increases and decreases in the coefficient, this indicated that the best
ARIMA model to fit the data was the integrated autoregressive moving average ARIMA
model.
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LAC 1 2345678 9
10 11
KEAN OF THE S E R I E S 6 2 . 1 3 8 9STD . DEV. OF S E R I E S 1 7 . 4 8 3 6NUMBER OF CASES 36
Figure 6. Partial Autocorrelation for the Hurgada Hotel.
Figure 6 presented partial autocorrelation plot; it showed the relationship among
specific lags holding all others constant. It indicated that the relationship between
different lags was a random one and the data could be analyzed using ARIMA models.
The autocorrelation test was performed to identify which ARIMA model best fit the data
for both hotels. The ARIMA models were estimated using unconditional ordinary least
square regression. The t-test and the goodness of t test (p-value) were used to test the
validity of the selected coefficients of ARIMA models.
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CORR. 0 . 3 5 9 0 . 1 4 1
- 0 . 0 7 4 - 0 . 0 8 3 ■ 0 . 3 0 7 ■ 0 . 2 0 7 0 . 1 1 9
- 0 . 0 9 1 0 . 1 6 9
- 0 . 0 8 0 0 . 0 1 6
• 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 u . o w . o i . . w
* * * * * * * *
• •* * * *
* • • *
• * *
CHAPTER 4
RESULTS AND DISCUSSION
The purpose of this study was to help provide the evidence of the efficiency and
accuracy of time series methodologies in forecasting hotels occupancy rates in Egypt.
Five time series models were used to fit time series data starting January 1991 until
December 1993 and to forecast six months ahead.
Characteristics of the Sample
A questionnaire was developed to collect monthly occupancy rate form two hotels
in Egypt. The results of the data collected reported that the forecasting process was
conducted by the sales, front office or reservation departments. The two hotels reported
that they did not use any quantitative methods in forecasting and that the forecasts were
prepared based on actual reservation figures adjusted by judgement. Both hotels were
different in size, type of operation and major market segmentation. The Cairo hotel was
a large city center hotel with 688 rooms, nine restaurants and three bars. The Hurgada
hotel was a resort hotel with 119 rooms, two restaurants and two bars. The major market
segment for Cairo was a mix of corporate, government and travel business. In the
Hurgada hotel travel was the major market segment which was due to its location in a
remote resort area.
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Time Series Models Fit
Brown’s model
The data was first fitted to Brown’s model in order to get the best fit. The best
fit is the one that had minimum sum of squared errors SSE. The value of a coefficient
was changed repeatedly until minimum SSE was reached. The model that had the best
fit was(cr = 0.67) for the Cairo hotel and (a = 0.48) for the Hurgada hotel. These were
the models used to forecast six months ahead.
Holt’s model
The data was fitted to Holt’s model in order to get the best fit. The best fit is the
one that had minimum sum of squared errors SSE. The value of a , B and parameters
was repetitively changed until minimum SSE was reached. The Holt's model that had the
best fit was (a = .67, 8 = 0.01) for the Cairo hotel and (a = 0.48, 8 = 0.01) for the
Hurgada hotel. The fitted models were used to forecast six months ahead.
W inter’s model
The data was fitted to Winter’s model in order to get the best fit. The best fit is
the one that had minimum sum of squared errors SSE. The value of a , 8 and 0
coefficient was repetitively changed until minimum SSE was reached. The model that had
the best fit was (o = .67, 8 = 01, 0 = .99) for the Cairo hotel and (a = 0.48, 8 =
0 .01, 0 * 0.99) for the Hurgada hotel. These model were used to forecast six months
ahead.
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ARIMA Model
The ARIMA model estimation was performed using ordinary least squares. The
coefficients were tested for significance using t-test and p-test. As presented in Table 1,
the ARIMA model for the data collected from the Cairo hotel proved to be ARIMA
(0,0,0,1). This means that only seasonal moving average (SMA) coefficient was used in
the model with a t-value of 39.56 and a p-value of 0.0000. The autoregressive (AR),
moving average (MA) and seasonal autoregressive (SAR) were not used in the model
because they proved not to be significantly different from zero.
Table 1. Unconditional Least Squares ARIMA Model for the Cairo Hotel
TERM COEFFICIENT STD ERROR T VALUE P VALUE
SMA 1 1.00985 0.02553 39.56 0.0000
Table 2 presents the result of ARIMA model for Hurgada hotel time series data,
the ARIMA model proved to be ARIMA (1,1.0.0). This means that only AR and MA
coefficients were used in this model. The parameters SMA and SAR were not used in
the model because they proved not to be significantly different form zero. The t-value
for AR was -3.29 and the p-value was 0.0010. For the MA t-value was 27.04 and p~
value was 0.0000.
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Table 2. Unconditional Least Squares ARIMA Model for the Hurgada Hotel
TERM COEFFICIENT STD ERROR T VALUE P VALUE
AR 1 - 0.47159 0.14320 -3.29 0.0010
MA 1 0.94184 0.03484 27.04 0.0000
Appendix B presents time series plots of actual occupancy rate versus the best fit
of different time series models used to forecast the occupancy for the Cairo hotel.
Appendix C presents a lime series plot of actual occupancy rate versus the best fit of
different time series models used to forecast the occupancy for the Hurgada hotel.
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Time Series Models Forecast
Six months of the collected data (January 1994 to June 1994), were reserved to
be compared with the forecast of the different models. The six months were forecasted
by different time series models. The difference between the actual occupancy and the
forecasted was measured using SSE test to measure the accuracy of forecasts. The most
accurate model in forecasting was the one that had minimum SSE for forecast.
Table 3 presents the results of the Cairo hotel and Table 4 presents the results of
the Hurgada hotel. Both tables presents actual six months occupancy rates from January
1994 untill June 1994 and the forecast of the five different time series models for the
same period.
The first objective of the study was successfully achieved as indicated in the
results presented in Table 3 and Table 4, the results are very close to the actual
occupancy rates. Most of the values were within 10 points deviation from the actual
occupancy rate. These deviations were within the acceptable range for the hotel industry
and considered very accurate compared to the existing judgmental forecasting errors in
hotels. The results presented in Table 3 and Table 4 indicated that the forecast of the
time series models was accurate and could be used to forecast hotels occupancy rates in
Egypt The time series models were able to smooth the random variations, trends, and
seasonality in the data to forecast six months occupancy rates with minimum deviations
from the actual occupancy figures.
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permission.
Table 3 Six Months Actual Occupancy versus the Forecast ofTiroe Series Models for the Cairo Hotel.
Month Actual
Occupancy
Models
ARIMA
F R
BROWN
F R
HOLT
F R
MVAV
F li
WINTRR
F R
Jan 72 50 59 50 13 00 58 40 14 10 61 93 10 57 54 35 18 15 58 55 13 95
Feb 45 70 58 00 -12 30 56 14 -1044 62 04 -16 34 52 88 -7.18 63.12 -17 42
Mar 52 00 56 50 i o 53 89 -1 89 62 41 -1041 5141 0 59 36.21 15.79
Apr 57 20 55 00 2 20 51 63 5.57 62 25 -5 05 49 94 7 26 64 44 -7 24
May. 61.40 53 50 7 90 49 38 1202 62.35 -095 48 47 12 93 60.90 0 50
Jun. 48 20 52 00 -3 80 47 13 1 07 62 16 -14 26 47 00 1 20 57 37 -9 17
F~ Forecast
F-a F.rror
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Table 4 Six Months Actual Occupancy versus the Forecast o f Time Series Models for the Hurgada Hotel
Month Actual Models
Occupancy ARIMA UROWN HOLT MVAV W IN! RR
F H F I F R F R F R
Jan 70 00 74.25 -4.25 64 29 5 71 65 68 4 32 52 30 17.70 43 03 26 97
Feb. 72 00 67 75 4 25 63 63 8 37 65 69 6 31 51 48 20 52 56.21 15.79
Mar 67 00 71.97 -4.97 62 98 4 02 65 71 1 29 5065 16 35 37.20 29.80
Apr 66 00 71 14 -5.14 62 33 3 67 65 72 0 28 49 83 16.17 61 44 4 56
May. 52 00 72 70 -20 70 61.69 -9 69 65.74 -13.74 49 00 3.00 43.73 8 27
Jun 470 0 73 12 -26 12 61 0 3 ■1403 65 75 -18 75 48 18 -1 18 54.74 -7.74
Fra Forecast
H“ Hrror
To address the second objective of the study, the accuracy of forecasts was
measured for each model using the SSE test. The results for the Cairo hotel are presented
in Table 5 and the results for the Hurgada hotel presented in Table 6. The most accurate
time series model to forecast the occupancy for the Cairo hotel was the ARIMA model.
It had the minimum SSE for forecasts which was (422.20). This result is consistent with
the findings of Andrew, Crange and Lee (1990) who conducted a similar study to
forecast hotel occupancy rate in a full service city center hotel using different time series
models. The authors reported that the ARIMA model was the best time series model to
forecast hotel occupancy rate.
The results were different for the Hurgada hotel. The most accurate model to
forecast the occupancy rates was Brown's exponential smoothing which had the minimum
SSE for forecasts equal to (423.00). This indicates that when changing the type of hotels
from a city center to resort hotel the ability of different time series models to forecast
the occupancy changed. This is because each hotel data is characterized with a certain
pattern that can be best captured using the specific model that can handle this specific
type of data.
43
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73-
ocU 66' p A N
59 •Y
RAT 5 2 iES
4 5 -
21 3 54 6
- A C T U A L
- A R IM A
- B R O W N
- M V A V
JANUARY 94 TO JUNE 94
Figure 7. The Forecast of the Best Models to Forecast for the Cairo Hotel
Figure 7 summarizes the forecasts of the best three models used to forecast the
occupancy for the Cairo hotel. Although the actual occupancy had random shocks
(unexpected fluctuations), the ARIMA, Brown and moving average time senes models
were able to forecast accurately. The most accurate model to forecast was the ARIMA
(SEE - 422.20), Brown’s exponential smoothing model came next and very close to
ARIMA (SEE=488.00). The third most accurate model was the moving average
(SEE=602.70).
44
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occupANCY
RATES
75
68
61
54
47
cw421 J
- a c t u a l
-• BR OW N
- HOLT
- ARIMA
JANUARY 94 TO JUNE 94
Figure 8. The Forecast of the Best Models to Forecast for the Hurgada Hotel
Figure 8 summarizes the forecast of the best three models used to forecast the
occupancy for the Hurgada hotel. The actual occupancy was decreasing exponentially.
The most accurate model to forecast for six months ahead was the Brown's exponential
smoothing (SEE=423.00). Holt’s exponential smoothing model was next (SEE=600.60),
which was followed by the ARIMA (SEE= 1198.00). Brown's model was the only model
to respond to the exponential decrease in the actual occupancy.
The results of this study indicate that the accuracy of time series models
(forecasting errors) change by changing the location, type of hotel operation and market
45
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segmentation. There is no single time series model that can be generalized to be used
based on its accuracy in forecasting hotel occupancy rates. The results suggest that
different models must be investigated for different types of hotels until the most accurate
model that fits a specific hotel is found. Then this model can be used to forecast the
occupancy rates for that particular hotel. This is because the patterns in the data for
different types hotel operations are different. The patterns can be best smoothed only by
using the specific model that was built to capture these patterns and use it to forecast the
future. This is supported by the findings of Makridakis (1986) who indicated that no
single method is superior for all accuracy measures and forecasting horizons.
HVAV BROWN BOLT W INTO ARIMA
Figure 9. SSE of Time Series Model Forecasts for the Cairo Hotel.
46
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Figure 9 indicated that the ARIMA model was the most accurate model to forecast
the occupancy rates for the Cairo hotel. Brown’s model was next which was followed by
the moving average model. The plot showed that although Brown’s model came second
in accuracy, it was very close in its prediction to the ARIMA model.
2500
2000 J
1500-i'
SSE
1000
500
MVAV BROWN HOLT WINTER ARIMA
Figure 10. SSE of Time Scries Model Forecasts for the Hurgada Hotel.
Figure 10 indicated that Brown’s model was the most accurate model to forecast
the occupancy rates for the Cairo hotel. Holt’s model was next which was followed by
the ARIMA model. The plot showed that although the ARIMA model came third in its
accuracy, the simple moving average model was very close to the performance of the
47
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ARIMA model in forecasting occupancy rates.
The results presented in Table 5, Table 6, Figure 7 Figure 8 Figure 9 and Figure
10 indicated that the accuracy of the most simple methods such as simple exponential
smoothing (Brown’s model), is comparable to or better than the most sophisticated time
series methods (Winter’s & ARIMA models). This is good news for the hospitality
industry because the simple methods used in this study are accurate and easy to develop.
This advantage makes it applicable in the area of forecasting for hotel occupancy rates
in Egypt. Hotels of any size can implement the simple time series models by using a
personal computer, simple statistical analyses software and minimum training for the
person preparing the forecast.
48
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Table 5. Sum of Squared Errors of Forecast (SEE) for the Cairo Hotel.
Moving average Brown Holt Winter ARIMA
602.70 488.00 716.80 884.14 422.20
Table 6. Sum of Squared Errors of Forecast (SEE) for the Hurgada Hotel.
Moving average Brown Holt Winter ARIMA
1274.00 423.00 600.60 2013.80 1198.00
49
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CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
For forecasting monthly occupancy rates in a selected sample of two hotels in
Egypt, the ARIMA model proved to be the most accurate model in forecasting six
months ahead in a full service city center hotel. In forecasting the same period for the
resort hotel, Brown’s exponential smoothing model outperformed all other time series
models.
In general, the fitted (moving average. Brown’s and Holt) simple time series
models performed close to or better than most sophisticated time series (ARIMA and
Winter’s) models. These results are significant in the fact that simple time series models
are easier to implement using very limited resources such as a personal computer and a
time series software to produce accurate forecasts for hotels.
The study indicates that the accuracy of forecasts varied for different types of
hotels. The more sophisticated method ARIMA. was more accurate in forecasting the
occupancy rate in a full service city center hotel, but did not perform as accurately for
a resort hotel. The time series simple models (Brown's and Holt’s) outperformed the
more sophisticated models (ARIMA) to forecast for the resort hotel. The findings of this
study indicate that the accuracy of different time series models vary by changing the hotel
type, location and market segment. This is due to changing patterns in the data of every
hotel. This will raise the question “which of the investigated models is the most accurate
in forecasting hotel occupancy rates ?’ . To answer to this question, the model that can
best smooth the variability and capture the trends of the time series data will be the one
50
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that most accurately forecasts the future. There is no specific model that can be used in
all forecasting situations. Different time series models must be investigated first to find
which is the most accurate model to forecast the data of a specific type of hotels. Once
the model is found it could be implemented to regularly forecast occupancy for that type
of hotels. Monitoring the model is very important to make sure that the patterns in the
data being forecasted are consistently repeated over the time and do not turn around in
a reverse trend because this will affect the ability of the model being used to forecast.
This study is significant because the simple time series models can be prepared
with a minimum data entry, minimum data collection (historical data of one variable) and
minimum training to prepare accurate forecasts and use it as a management tool for better
planing in all ares of the hospitality industry. The empirical evidence provided in this
study on the accuracy of time series applications in the hotel industry should be given a
serious consideration by the hospitality educators. It might be useful to adopt this method
of forecasting due to its importance and cost effectiveness as a management tool in the
hospitality industry.
Future research should be focused on investigating the monthly occupancy rates
when disaggregated into its components by weekly occupancy. Weekly occupancy rates
should then be forecasted and aggregated to re-establish the monthly occupancy rates.
This technique could be applied to determine weather or not there will be an increase in
the accuracy of time series models. In another area for future research, it would be
valuable to investigate the occupancy time series data when disaggregated into its
components by market segments (conference, walk in, day use, travel). Each segment
51
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should be forecasted using time series models and then aggregated to form the forecasted
occupancy rates. This technique could enhance the accuracy of the time series forecasting
models.
The time series models used in this study were only investigated in the area of
occupancy forecasting. For future research, one might apply the time series models in
different areas of the hotel to investigate their accuracy in forecasting other variables like
inventory levels, table turnover in the food and beverage outlets, material requirement
planning for food production, and forecasting power needs such as energy usage in
hotels.
52
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References
Andrew, W. P., Crange, D., & Lee, K. (1990). Forecasting hotel occupancy rates with time series models: An empirical analysis. Hospitality Research Journal. 14,(2), 93-127.
Adam, E. E. and Ebert, R. J. (1976). A comparison of human and statistical forecasting.A l H t l a n s . , S. 120-127.
Armstrong, J. (1983). Relative Accuracy of judgmental and extrapolation methods in forecasting annual earnings. Journal of Forecasting. 2, 437-447.
Box, G. E. P. and Jenkins G. M. (1976). Time series analysis forecasting and control (rev. ed.). San Francisco, CA: Holden-Day.
Binroth, W., Burshtein I., Haboush, R. Sc Hartz J. (1979). A comparison of commodity price forecasting by Box-Jenkins and Regression - based techniques Technological Forecasting and Sociai Change. ]4 , 169-180.
Cerullo, M. J. and Avila, A. (1975). Sales forecasting practices-a survey. Managerial Planning. September/October, 33-39.
Christ, C. F. (1951). A test of an Econometric model of the United States. 1921-1974. In Conference on Business Cycles. New York: National Bureau of Economic Research.
Christ, C. F. (1975). Judging the performance of econometric models of the U.S. economy. International Economic Review. 1£,(1), 57-81.
Clearly, J. P. and Fryk, D. A. (1974). A comparison of ARIMA and econometric models for telephone demand. Proceedings of American Statistical Association. Business and Economic Section, 448-450.
Cleveland, W. P. and Tiao, G. C., (1976). Decomposition of seasonal time series: A model for the census X-H program. Journal of American Statistical Association. 71. 581-587.
Cooper, R. L. (1972). The predictive performance of quarterly econometric models of the U.S. Conference on Econometric Models of Cyclical Behavior. New York: National Bureau of Economic Research.
53
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Cooper, J. P. and Jergenson, D. W. (1969). The predictive performance of Quarterly econometric models of the U.S. Conference on Econometric Models of Cyclical Behavior, Harvard University, November, 14-15.
Cooper, J. P. and Nelson, C. R. (1975). The ex ante performance of the St. Louis and FRB-MIT-PENN econometric models and some results on composite predictors. Journal of Money. Credit and Banking. February, 1-32.
Crange, D. and Andrew, P. (1992). A comparison of time series and economic models for forecasting restaurant sales. International Journal of Hospitality Management. 11.(2), 129-143.
Dalrymple, D. J. (1975). Sales forecasting methods and accuracy. Business Horizons. 5, 69-74.
Geurts, M. D. and Kelley, J. P. (1986). Forecasting retail sales using alternative models. International Journal of Forecasting. 2. 261-272.
Geurts. M. D. and Ibrahim, I. B. (1975). Comparing the Box-Jenkins approach with the exponentially smoothed forecasting model application to Hawan tourists. Journal of Marketing Research. 12. 182-188.
Granger. C. W. J. (1969). Investigating causal relations by econometric models and cross spectral methods. Economctrica. 32 , 424-438.
Groff, G. K. (1973). Empirical comparison of models for short range forecasting. Management Science. 200). 22-31.
Gross, D. and Ray, J. L. (1965). A general purpose forecasting simulator.Management Science. fiU(6). 119-135.
Hollier, R. H., Khir, M. & Storey, R. R. (1981). A comparison of short-term adaptive forecasting methods. OMEGEA International Journal of Management Science. 2. 96-98.
Hogarth, R. M., (1975). Cognitive processes and the assessment of subjective probability distributions. Journal of The American Statistical Association. 7Q, 271-290.
Kirby, R. M. (1966). A comparison of short and medium range statistical forecasting methods. Management Science. B15(4), 202-210.
Krampf. R. F. (1972). The turning point problem in smoothing models. Unpublished Ph.D. thesis. University of Cincinnati.
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Lorek, K. S., McDonald, C. L. & Patz, D. H. (1976). A comparative examination of management forecasts and Box-jenldns forecasts of earnings. Accounting Review. 2, 321-330.
Levine, A. H. (1967). Forecasting techniques. Management Accounting. January, 86-95.
Libby, R. (1976). Man versus model of man: some conflicting evidence. Organizational Behavior and Human Performance. 4, 1-12 .
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McWhorter, A. Jr. (1975). Time series forecasting using the kalman filter: An empirical study. Proceedings of the American Statistical Association. Business and Economic Section, 436-446.
Makridakis, S. and Hibon, M. (1979). Accuracy of forecasting: An empirical investigation. The Journal of the Roval Statistical Society. 2, 97-145.
Makridakis, S. (1986). The art and science of forecasting: an assessment and future directions techniques. International Journal of Forecasting. 2. 15-39.
Makridakis, S. and Wheelwright S. (1989). Forecasting Methods for Management. New York, John Wiley & Sons.
Maibert, V. A. (1975). Statistical versus sales force-executive opinion short-range forecasts: a time series analysis case study. Krannert Graduate School, Perdue University (working paper).
Narashimhan, G. V. L. (1975). A comparison of predictive performance of alternative forecasting techniques: time series models vs. an econometric model. Proceedings of the American Statistical Association. 1, 459-464.
Nelson, C. R., (1972). The prediction performance of the FRR-MIT PENN model of the U.S. Economy, American Economic Review. 62. December, 902-917.
Newbold, P. and Greanger, C.W.J. (1974). Experience with forecasting univariate time series and the combination of forecasts. Journal of Roval statistical society. 122(2). 131-165.
Nylor, T. H. and Seaks, T.G. (1972). Box-Jenldns methods: an alternative to econometric models. International Statistical Review. 40(2). 113-137.
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Poulos, L., Kvanli, A. and Pavur, R. (1987). A comparison of the Box-Jenkins method with that of automated forecasting methods. International Journal of Forecasting. 2, 261-267.
Raine, J. E. (1971). Self-adaptive forecasting considered., Decision Science. 2(2), 264- 279.
Reid, D. J. (1969). A comparative study of time series prediction techniques on economic data. Unpublished Ph.D. thesis, University of Nottingham.
Reid, D. J. (1971). Forecasting in action; comparison of forecasting techniques in economic time series. Proceedings of the Joint Conference of Operation Research Society.
Reid, D. J. (1975). A review of short term projection technique. Practical Aspects of Forecasting, Operational Research Society. London. 6, 8-25.
Schmidt, J. (1979). Forecasting state retail sales: Econometric versus time series models. The Annuals of Regional Science. 8. 91-101.
Simon, H. and Newell A., (1971). Human problem solving: The state of the theory in 1970. American Psychologist.28. 1-39.
Steckler, H. O. (1968). Forecasting with econometric models: an evaluation. Econometnca. 26. 437-463.
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a p p e n d ix e s
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APPENDIX A
Sample of the questionnaire used for data collection
S8
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Predicting Future occupancy Rates in Hotels Using
Forecasting Models
1) Name ..................................................
Title..................................................
A d d r e s s .......................................... C i t y .........
Country ......................................................
2) Hotel Name...........................................................
Address.........................................
C i t y .........................................................
Country.........................
3) Number of Rooms...................
4) Number of Restaurants..........................
5) Number of Bars....................................
6) How is your hotel managed ? Please circle one only.
a) Owner b) Management Company c) Franchise
d) Other, please s p e c i f y ...........................................
7) If the hotel was not managed by the owner, then what is the
name of the firm managing the hotel ?
59
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8) What is the major market segment of the hotel ?
Please circle one segment only.
a) Corporate b) Resort c) Travel d) Government
e) Conference f) Transient
g) Other, please specify
9) Do you use computer software in occupancy forecasting
Yes No
10) If yes, please list the name of the software used.
11) Please circle the type of computer hardware used in
forecasting.
12) If a personal computer is used for forecasting, is it:
a) a Stand alone system (just for forecasting) or,
b) an integrated system with hotel software applications such
as front office, yield management, etc.
13) If you do not use computers, please circle one cr more of the
following methods used for occupancy forecasting.
a) Quantitative methods b) Judgment based on experience
c) Sales people reports
d) Other please specify ...................................
a) Personal computer b) Main frame
60Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14) If quantitative methods are used in forecasting what are the
methods used. (Please circle one or more).
a) Simple average b) time series
c) Econometric methods
d) Other, please specify....................................
15) How often do you forecast the occupancy ? (Please circle one or more of the following).
a) Daily b) Weekly c) Monthly
d) Quarterly e) every 6 months f) yearly
g) Other, please specify...........................................
16) The forecasting process is done by: (Please circle one or more
of the following)
a) Sales person b) Sales manager
c) Marketing manager d) General Managere) Other, please specify .........................................
17) Please complete the attached tables for the historical monthly
occupancy percentages; starting January 1989 thru June 1994,
and list vour forecast from July,1994 till December 1994 in
the attached tables.
61
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permission.
Occupancy R ates (1991)M k ( Sgmnt JJ*1! D t f c t e
R ack Rale
T rav e l A tcn l .
Conference
T rans ien t
C roups
C ovenm cnl
C orpo ra teHi
oo muF§lit R n w P i n SHI mhm M ' k ' mWM
D e c : : ‘!
1
Total M onthly Occupancy
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ner. Further
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without
permission.
O ccupancy Rates (1992)M kt Sgmnt 5 y 6 8 ^ Nov'«> h c c ; f e ;
R ack R ale
T ra v e l A fen l.
Conference
T ra m le n l.
C r o u p i
Covcnm enl
C o rp o ra te m Ml m md& T,*Nov.' i> 'te =
1
—— ■■ 1 \
Tnlal Monllily Occupancy
Occ
upan
cy
Rat
es
(199
3)
•k
uV.
JC
64
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O ccupancy Hales In Percent (1994)Forecast Foreca st Forecast Forecast Forecast Forecast
M arket Segm ent 1 Jan. ' Feb: WAr. Apr.’ May.’ \ ' w "■June '■Sept"; i ' O t U - Nov'. . Dec.;Rack RaleT ra w l Accnl.ConferenceT ra n i ic n lG ro u p iC o w n m e n fC o ipo ra lc
other V fl|aH U»tJJfl*W' Occupancy/Mk| Sfmal '
4* v . Jan. Feb. f>Ur.
<
Apr. May.***■ Forecast Forecast Forecast Forecast Forecast Forecast
’June Ju ly Aue. Sept. Oct. N ov . Dec.
Tolal M onthly Occupancy
APPENDIX B
Time scries plots of models fitting for Cairo Hotel
66
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CAllt
O (K
.'CU
PAN
CY
KA
'IIO
S
APPENDIX B.l
5 0 -
3 0 -
4 32 9 3 68 22l 1 5
JANUARY 9 1 TO DECEMBER 9 3
Figure II. Moving Average Model Fit for the Cairo Hotel.
• A c tu a l
• Forecast
67
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CAIR
O OC
CUPA
NCY
RA'I»
>
APPENDIX B.2
90 4
70i
433629228 15Case Number
A lp h a “ 0 6 7
Figure 12. Brown’s Model Fit for the Cairo Hotel.
• A c tu a l
• Forecast
68
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APPENDIX B.3
9 0<
>
7 0 -
SO
3 0
2 9 3 6 4 3228 151Case Number
A lp h a - 0 . 6 7 B a ta ■ 0 0 1
Figure 13. Holt’s Model Fit for the Cairo Hotel.
69
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• A c tu a l
• P o ra c a s t
CAIK
O (X
X’U
I’ANC
Y ItA
'IKS
APPENDIX B.4
9 0 -
70
F o ra c a s t
22 29 368 431 15Case Number
Alpha « 0 67 B a ta - 0.01 G a m m a - 0 .99
Figure 14. Winter’s Model Fit for the Cairo Hotel.
70
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APPENDIX B.5
<
>z<
5<9 50-<* '
3081 22 29 3615 4 3
Case Number
Figure 15. ARIMA Model Fit for the Cairo Hotel.
71
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A c tu a l
F o r a c a s t
APPENDIX C
Time scries plots of models fitting for Hurgada Hotel
72
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APPENDIX C .l
£<
%<<*
100 -
6 0 -
4 0 -
20 ■ —
8 IS 22 29
JANUARY 9 1 TO DECEMBER 9 3
36 43
Figure 16. Moving Average Model Fit for the Hurgada Hotel.
73
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Actual
Forecast
APPENDIX C.2
too •
>
6 0 t
X<<
29 361 8 1 5 22 43
Case Number Alpha - 0 .48
Figure 17. Brown’s Model Fit for the Hurgada Hotel.
74
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A c tu a l
For«c«t
APPENDIX C.3
£< 80-1>Z<
p 60-P
<<
298 362215 431Case Number
Alpha - 0 .4 8 Beta - 0 01
Figure 18. Holt’s Model Fit for the Hurgada Hotel.
75
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Actual
F o r e c a s t
IIUI
iGAD
A (X
X.'U
PANC
Y R
A'I
KS
APPENDIX C.4
100 ■
60 ■
4 0 -
36 4 3298 22151Case Number
Alpha - 0 .4 8 Bata - 0.01 Gamm a - 0 .99
Figure 19. Winter’s Model Fit for the Hurgada Hotel.
• Actual
• Forecast
76
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APPENDIX C.5
100 ■
<
>5<
>rx<<
36 4329228 151Case Number
Figure 20. ARIMA Model Fit for the Hurgada Hotel.
77
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Actual
F o r e c a s t