development of a demand forecasting model for a …
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DEVELOPMENT OF A DEMAND FORECASTING MODEL FOR
A SUMMER FOOD SERVICE PROGRAM SPONSORED BY
THE UNITED STATES DEPARTMENT OF AGRICULTURE
by
MICHELLE MIN6-HSUEH LIN, B.S.
A THESIS
IN
RESTAURANT, HOTEL, AND INSTITUTIONAL MANAGEMENT
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCE
Approved
December, 1995
£0S ^ /'' "f^^^ J Z, ^ .- ACKNOWLEDGEMENTS ' / ' ^
C&:}-^1- This thesis is the result of a collaborative effort.
Although I labored very diligently to complete this
research, I readily concede this undertaking would not have
been possible without the advice and help I received from
others. It is these individuals, whose support and
assistance was so generously given, that I acknowledge and
to whom I give thanks. Dr. Linda C. Hoover has been
tremendously helpful in the professional advice and support
she has so freely given to me, and I express thanks to Dr.
Julia T. Poynter for her valuable assistance, also. I
sincerely appreciate the kind support I received as well
from Dr. Ronald H. Bremer, whose advice in helping me with
data analysis was indispensable. For her gracious
assistance in giving me access to the Summer Food Service
Program data I used in this study, I convey my sincere
gratitude to Debora Phillips.
With their limitless love and support, my family in
Taiwan also helped me to complete my Master's degree. I
take special pride in dedicating this thesis to my late
father, whose encouragement motivated me to achieve my goal.
11
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
LIST OF TABLES vi
LIST OF FIGURES vii
CHAPTER
I. INTRODUCTION 1
Background 1
Statement of the Problem 6
Purpose of the Study 6
Research Questions 7
Definitions 8
Research Assumptions 10
Research Limitations 11
Contribution of Proposed Study 11
II. REVIEW OF LITERATURE 12
Forecasting in Food Service Operations . . . 12
Commercial Operations 12 Non-commercial Operations 13
Importance of Forecasting: The Impact on Food Production Management 15
Controlling Labor Costs 15 Controlling Food Costs 16 Maximizing Customer Satisfaction . . . . 16
Elements of Food Service Forecasting . . . . 16
Historical Records 16 Pattern of Demand 17 Forecasting Time Periods 19
Demand Forecasting Models 21
Types of Forecast Methods 21
iii
Criteria for Forecasting Models . . . . 35 Measuring the Accuracy of Forecast
Models 39
III. METHODOLOGY 43
Pilot Study 43
Sites 46
Data Collection 46
Treatment of Data 47
Data Analysis 47
Phase I--Selection of an Appropriate Forecast Model 47
Phase II--Evaluation of Menu and Daily Indices on the Accuracy of a Forecast Model 54
IV. RESULTS AND DISCUSSION 57
Pattern of Time Plots 57
Accuracy of Forecast Models 59
The Most Appropriate Forecast Method . . . . 63
Alpha Analysis 66
Daily Index 70
Menu Index 73
Recommendation to SFSP in West Texas . . . . 79
Comparison of Results 79
V. CONCLUSIONS 82
Major Findings of the Study 82
Impact of the Study 83
Limitation of the Results 85
Recommendations for Further Research . . . . 86
Summary 86
IV
REFERENCES 89
APPENDICES
A. SUMMER FOOD SERVICE PROGRAM FOOD PRODUCTION RECORD 94
B. DATA ANALYSIS SPREADSHEET 95
C. ESTIMATED NUMBER OF MEALS FOR PREPARATION . . 96
D. SELECTED COLUMNS OF RAW DATA AND TIME PLOTS . 97
V
LIST OF TABLES
1. Ranking values of forecast methods based on objective and subjective analysis for pattern of time plot. 51
2. Example for one site: Ranking values for accuracy of forecast methods based on MAD, MAPE, and RMSE . 53
3. Example for one site: Selection of an appropriate forecast method based on pattern, accuracy, and simplicity 53
4. Trend analysis of time plots for meals served at SFSP sties 58
5. Seasonality analysis of time plots for meals served at SFSP sites 58
6. Analysis of RMSE by forecast method for meals served at SFSP sites 60
7. Analysis of MAD by forecast method for meals served at SFSP sites 61
8. Analysis of MAPE by forecast method for meals served at SFSP sites 62
9. Comparison of mean ranking score for appropriateness of the forecast methods for breakfast and lunch 67
10. Comparison of mean alpha for different patterns of time plot for breakfast and lunch 69
11. Forecast error measures with adjustment of daily index for breakfast 71
12. Forecast error measures with adjustment of daily index for lunch 72
13. Breakfast menu index analysis for selected sites of SFSP 76
14. Lunch menu index analysis for selected sites of SFSP 78
VI
LIST OF FIGURES
1. Final selection of forecast method for breakfast . 64
2. Final selection of forecast method for lunch . . . 65
3. Alpha distribution for simple exponential smoothing method 68
4 . Menu index analysis of breakfast 74
5 . Menu index analysis of lunch 77
Vll
CHAPTER I
INTRODUCTION
Background
In the very competitive and dynamic environment that
most businesses face, forecasting is a beneficial tool and
an indispensable strategy for business survival. Success in
analyzing and forecasting customer demand for a given good
or service can mean the difference between profit or loss
for an accounting period and, ultimately, the success or
failure of the business itself. Chandler and Trone (1982)
noted that forecasting is the starting point for all
budgeting. Forecasting also is used to predict daily sales,
utilizing past data which is adjusted for factors such as
management judgement, economic considerations, and current
trends.
The main purpose of forecasting is to predict future
events that could potentially affect the success of an
operation. All facets of management rely on the estimates
and predictions developed through forecasting. Therefore,
the primary focus of forecasting is to determine customer
demand for an organization's goods or services (Webster,
1986) .
Forecasting in the food service industry is invaluable
to various aspects of operations. The food service manager
needs to forecast sales in order to plan staff schedules and
make food and supply purchases. Over-forecasting means that
the demand is less than the forecast, which results in
wasted resources. On the other hand, under-forecasting will
result in employee stress and customer dissatisfaction.
Thus an accurate forecast is the goal of a food service
manager who strives to achieve a successful business
(Messersmith & Miller, 1991).
Food service operations can be classified as either a
commercial or non-commercial. They also can be classified
as either for-profit or non-profit. Regardless of the
nature of the food service operation, all food service
managers operate within fairly clear financial limits.
Therefore, any technique that can help to improve
operational efficiency by getting a more accurate picture of
demand and by limiting waste would be extremely helpful.
Many federal funded food programs are non-commercial/
non-profit operations. Cost control is extremely important
in this type of operation. The Summer Food Service Program
(SFSP), a Child Nutrition Program funded by the United
States Department of Agriculture (USDA), is such a program.
The purpose of the SFSP is to provide children with
nutritious food during the summer when school is not in
session. As a federally funded program, with limited
funding and non-profit attributes, SFSP has a critical need
for forecasting.
In Texas, the SFSP is administered by the Texas
Department of Human Services (TDHS) Special Nutrition
Program (SNP). There were 254 agencies involved in
operating the SFSP in Texas in 1994. Although more than 47%
of the 3.5 million Texas school children qualified to
receive benefit from this Child Nutrition Program, only
about 9.2% of those children actually received summer meals.
The SFSP provided 7.4 million meals to children in 1994
(Summer Food Service Program: Orientation & Organizing
Guide, 1995) . The successful operation of the SFSP will
bring more opportunities for children to receive nutritional
meals in the summer time.
Children's Enterprises Incorporated (CEI), a private
non-profit organization, currently operates the SFSP
throughout a vast geographic area in west Texas. CEI has
operated 10 to 15 cafeterias in low-income areas in this
area each summer since 1989. Through this program, free
breakfast and lunch meals are provided daily for children 18
years old and younger. Children and youth are not charged
for meals and do not complete any paperwork. They may
receive second servings, if they desire. The menu is
standardized in an eleven-day cycle, but substitutions occur
to provide for better inventory usage. The breakfast menu
must include a serving of milk, fruit, and bread or cereal.
The lunch menu must include a serving of meat, bread, two
4
fruits and/or vegetables, and milk (Summer Food Service
Program Handbook, 1995).
The locations of the CEI cafeterias are usually in
schools, but they also may be located at local youth
agencies or churches. CEI employs cafeteria workers from
local schools to operate the meal service. This provides
employment opportunities for school personnel during summer
months and benefits CEI's SFSP by utilizing experienced
employees and established kitchen facilities. The service
period for the SFSP is usually from the week after the
regular school year ends until two weeks to one month prior
to the beginning of the regular school year. Most sites
operate for six to twelve weeks during the summer (Summer
Food Service Program: Orientation & Organizing Guide,
1995) .
Food inventory must be consumed by the end of the
summer for three reasons. One reason is the distance
involved in travelling from sites to the administrative
office, which can be up to 180 miles. A second reason for
eliminating the inventory is that the administrative office
has limited space to store these supplies. Finally, because
of the short shelf life and spoilage of some of the items,
they would not be usable for the following year's program
(Phillips, personal communication, October 15, 1994).
The SFSP is an example of a food service operation that
could benefit greatly from forecasting techniques. The
administrator and staff of CEI have expressed a need for a
more precise forecasting method but no literature was found
to assist in choosing a forecast method for SFSP. Plans for
staffing and purchasing have been made by the "best guess"
method. Each year program plans are made based on the
number of children that were served during the previous
year. When a new cafeteria is opened, the forecast is made
based on past experience with other communities of similar
size. No mathematical methods have been applied to this
operation.
The initial supplies and groceries for each site are
purchased by the Lubbock administrative staff at the
beginning of the summer. Later, food service managers from
each cafeteria prepare weekly grocery orders based on their
judgement and previous experience and submit the orders to
the administrative office. This procedure lacks any
scientific basis. These orders may be modified by the
administrative staff, again, based on judgement and previous
experience and not mathematical methods.
Current operations do not incorporate a forecasting
method because of the following reasons: (1) the staff at
each cafeteria site lack forecasting skills, (2) the program
does not have a forecasting system available for the staff
at each site to follow, (3) the program runs in a very short
time span, and training time is limited, and (4) since
employees may change from one summer to the other, the
program is not willing to train the staff in how to use a
complex forecasting technique. It is not practical for the
administrative staff in Lubbock to calculate daily forecasts
since this would require at least two long distance calls
per site each day.
Statement of the Problem
The SFSP in west Texas, utilizing the "best guest"
forecast method, has confronted major operational problems
and has a great need to implement a forecasting technique to
solve this problem. Therefore, finding an accurate and
efficient forecasting method is very important for this
operation.
Purpose of the Study
The purpose of this study was to explore the
application of appropriate forecasting methods to an
existing food service operation. This study compared
different models of forecasting and selected an appropriate
forecasting method based on three criteria: the pattern of
demand, the accuracy, and the simplicity of the model. The
specific objectives of this study were to:
1. screen patterns of time plots of each site to
determine if there was a trend or seasonality in
the past data.
2. compare the accuracy of various forecast models
for both individuals and aggregate data,
3. analyze the procedures required for each forecast
model being tested and rate its simplicity,
4. recommend the best forecast model for the SFSP
based on pattern, accuracy, and simplicity, and
5. determine if the menu item or the day of the week
affects demand, and
6. create a worksheet which allows food service
operations to apply the forecast method
recommended as a result of the study.
The forecasting method selected must be a simple one
that does not rely on computer resources or extensive
training, in order to keep the cost of training and hardware
investment minimal. As a result of this study, CEI will
have the ability to plan, purchase, and staff more
efficiently.
Research Ouestions
The research questions to be addressed by this study
were:
1. What is the menu preference of the SFSP in the
past three years?
2. What is the demand trend along the operational
period?
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3. What is the most accurate forecast model for each
CEI cafeteria site for breakfast and lunch?
4 . What is the most accurate forecast method for the
aggregate data of the combined cafeteria settings?
5. Based on the calculation procedure required for
each testing forecast models, what is the simplest
forecast method?
6. Based on findings from questions 2, 3, 4, and 5,
what is the best method for demand forecasting to
achieve more efficient and effective planning of
staffing and purchasing for the SFSP in west
Texas?
7. Does the menu item and demand trend affect the
accuracy of forecasting?
Definitions
"Best Guess" Method--The current forecast method used
by the staff of the SFSP operated by CEI's is based on no
mathematical forecasting method. The estimation of
forecasting is based on the forecaster's previous experience
and intuition (Phillips, personal communication, October 15,
1994) .
Daily Index--Daily index is the ratio of the total
servings from one weekday (for example, Monday) to the total
servings from the whole week (Wheelwright & Makridakis,
1985) .
Demand--The desire to purchase a good or service
(Nisberg, 1988). The demand mentioned in this study is the
actual serving count for each meal of the SFSP.
Demand Trend--Demand trend identifies the gradual
increase or decrease in demand (Wheelwright & Makridakis,
1985) .
Forecasting--Spears (1991) explained forecasting as the
art and science of estimating future events by combining
intuitive interpretation of data with the use of
mathematical models. The primary purpose of this study is
to predict the meal service count for each meal of the SFSP
operated by CEI in west Texas.
Forecast Model--The forecast model is the technique
that either utilizes a mathematical or non-mathematical
methods to estimate the forecast (Chase & Aquilano, 1992) .
Mathematical Forecasting--A quantitative forecasting
technique that requires a certain formula to calculate the
forecast demand (Wheelwright & Makridakis, 1985).
Menu Preference Index--This index is the proportion or
percentage of servings of one menu item to the total
servings (Messersmith & Miller, 1991).
Over-forecasting--The estimation of forecasting is
higher than the actual demand (Messersmith & Miller, 1991) .
Pattern of Demand--The pattern of demand is the trend,
cycle, or seasonality appeared on the time plot (Chase &
Aquilano, 1992).
10
Scatter Plot--A scatter plot displays a statistical
relationship between two metric-variables (Cryer & Miller,
1991).
Seasonality--A pattern of demand occurs routinely in
certain intervals of time (Wheelwright & Makridakis, 1985).
Time Series--A time series is a series of measurements
taken at successive points in time (Iman & Conover, 1989).
Time Plot--A time plot is a plot of the time series
values versus time with successive points connected (Bremer,
personal communication, June 16, 1995).
Under-forecasting--The forecasting is under estimated
which leads to running out of food items for the meal
service (Messersmith & Miller, 1991).
Research Assumptions
The assumptions for this research were:
1. The economic base of the community do not affect
the demand patterns.
2. The population in the city where each site is
located will stay the same in the long term.
3. One site in the program has the same attributes as
another in considering demand patterns.
4. The demand for July 4th would be the same forecast
value as calculated by using simple exponential
smoothing method. July 4th is excluded as a
workday because it is a national holiday.
11
Research Limitations
The ability to generalize the results of this study was
limited by the following constraints:
1. This research analyzed only one contractor of the
SFSP in the state of Texas.
2. The sites under study were suburban and rural
locations. Since urban sites were not included,
the results might not be applicable to these
environments.
Contribution of Proposed Study
The food service industry is currently undergoing
tremendous change as the cost of operations continues to
rapidly climb for both non-commercial and commercial
operations. Because of the strong pressure on operating
margins and the need to control expenses and stay within
budgets, forecasting can be a tremendous benefit to both
large and small food service operations. Despite the strong
need for forecasting in food service, the application of
forecasting models is quite limited. Forecasting is
critical for the SFSP due to its limited funding and short
time span. This study identified a simple but realistic
demand forecast model for the SFSP, that may well be
applicable for other short time frame food service
operations.
CHAPTER II
REVIEW OF LITERATURE
Forecasting in Food Service Operations
Commercial Operations
Restaurants
Forecasting demand for goods and services is critical
for effective and efficient restaurant operations. Accurate
forecasting results in effective cost control which assists
profitability in restaurant operations. Despite the
benefits of using a forecasting method, recent research
(Repko & Miller, 1990) has revealed that few food service
operations use forecasting. Restaurant operators, however,
report a need for improvement in both training and
application of forecasting methods (Repko & Miller, 1990) .
The review of literature revealed that food service managers
do not use forecasting techniques because they do not fully
understand how to use them.
Airline Food Service
Forecasting of airline meals is based on the number of
passengers. Pedrick, Babakus, and Richardson's (1993) study
found that airline customers comment that the number of in
flight airline meals is generally underestimated. Because
of the time constraints in estimating an accurate count of
passengers, forecasting the number of meals needed on a
12
13
flight is difficult. Also, considerations of meal variety
and special meal requirements add to the difficulty of
forecasting.
Non-commercial Operations
Health Care Facilities Food Service
In the past, health care operators did not pay adequate
attention to forecasting primarily because they did not have
a strong economic incentive to do so. Costs associated with
inadequate forecasting were passed on to their customers.
However Reyna, Kwong, and Li (1991) stated that under a
third-party reimbursement system, payments are based on
amounts set by the government for each service. This change
in the payment method, plus rising costs and increasing
competition, have made hospital operations more budget
conscious. Health care managers are now looking for ways to
cut costs. Therefore, forecasting has become an essential
part of health care food service management.
College and University Food Service
Forecasting is especially important in college and
university food service operations which are usually non
profit. Repko and Miller (1990) conducted a survey in 1990
to assess the need for current application of forecasting in
college and university food service operations. The study
revealed that 79% of the respondents valued forecasting as
14
very important. Respondents also indicated a need for
improvement in training and application in the area of
forecasting. A similar study was conducted by Miller and
Shanklin (1988b). In their research, educators responded
that forecasting was an important tool for managers of food
service operations and that continuing training was
necessary in this area.
Child Nutrition Programs
The Summer Food Service Program (SFSP), funded by the
United States Department of Agriculture (USDA), was created
by Public Law 90-302 in 1968. This law was amended in 1975
under Public Law 94-105. The purpose of the SFSP is to
provide children with a nutritious meal during the summer
months when they are out of school and would not normally
receive the free or reduced price meals. The program was
initially called the Special Food Service Programs for
Children.
SFSP operations throughout the country are usually
operated by school districts, city parks and recreation
departments, or other non-profit organizations. Many sites
offer summer school, workshops or field trips for children.
For example, in 1989, Columbia, Missouri, offered a summer
program for children that included crafts, health and
nutrition education, swimming, sports activities, and field
trips in addition to the SFSP. The summer program provided
15
physical activities as well as nutritional meals for
children during the summer (Gibson, 1989; Ott, 1978; Summer
Feeding, 1978) .
The regulations for SFSP has become more and more
restricted. The regulations in 1995 ("Summer Food Service
Program", 1995) state that only lower income neighborhoods
may participate in the SFSP or participants must qualify for
a minimum income level. Current law defines the low income
area as an area in which one-half or more of the children
are from families with income at or below 185% of poverty.
This regulation was implemented to prevent program abuses,
but also has resulted in fewer sites qualifying for the
program. Increasing restrictions to operate a SFSP means
that the forecasting is critical for the success of
operating the SFSP (Summer Food Program Restricted, 1991) .
Importance of Forecasting: The Impact on Food Production Management
Controlling Labor Costs
Accurate forecasting of customer demands is critical
for realizing effective labor cost control. Pavesic (1983)
and Wacker (1985) indicated that accurate forecasting is one
of the prerequisites for labor cost control. Managers
schedule labor according to forecasts. Thus, accurate
forecasts can result in cost effective scheduling.
16
Controlling Food Costs
Over estimating demand (over-forecasting) leads to
overproduction and results in extra costs. Messersmith and
Miller (1991) stated that the problem with over-forecasting
is the cost of unused prepared food which includes labor
associated with handling, such as wrapping, storing,
recording, and replanning. Rehandling and discarding menu
items are also hidden costs of over-forecasting.
Maximizing Customer Satisfaction
Under-forecasting leads to under production. This can
result in customer dissatisfaction if they do not receive
their menu choices. The cost of under-forecasting may be
minor, but the cost of losing customers is significant.
Underproduction also can cause high stress for cooks,
service employees, and managers (Messersmith & Miller,
1991) .
Elements of Food Service Forecasting
Historical Records
Historical records are the most important element in
forecasting. As stated previously, forecasting is the
prediction of a future event based on past data. Therefore,
complete data and information are required in order to
forecast effectively. Spears (1991) indicated that
reliable forecasting depends on accurate and complete
17
records. The better the data available to a forecaster, the
more accurate the forecast will be. Therefore, it is the
responsibility of the forecaster to get as much historical
data and current information as possible before making a
forecast.
Pattern of Demand
The main task in forecasting is to analyze past data to
predict a future event. The forecaster must consider the
pattern of past data when making projections. The easiest
way to produce a forecast is to compute the average past
demand and use it to estimate the future demand. The
pattern of demand includes random variation, trend lines,
seasonal influence, and cyclical elements (Chase & Aquilano,
1992) .
Random Variation
Effective forecasting assumes a regular predictable
pattern of demand can be accurately determined. When random
variations occur, mostly caused by chance events, the
predictive power of forecasting is greatly weakened.
Forecasters are challenged by this type of demand. Random
events that can disrupt forecasting include strikes,
earthquakes, wars, and changes in weather (Jarrett, 1991).
Having a complete data record is very important to making an
effective forecast. Chandler and Trone (1982) indicated
18
that having a complete data record the first 12 months of
operations is most crucial for small businesses.
Trend Line
According to Chase and Aquilano (1992), four types of
trend line demand distributions are: linear trend, S-curve
trend, asymptotic trend, and exponential trend. Demand
distribution in a linear trend or horizontal pattern
indicates a straight continuous relationship. Schonberger
and Knod (1994) stated that trend lines define a positive or
negative shift in series value over a certain time period.
A straight line demand distribution shows stable sales.
Chase and Aquilano (1992) noted that S-curve trends indicate
the demand of the service through the stages of development,
growth, and maturity. Asymptotic trends indicate the
highest demand growth at the beginning of the service period
and then taper off. Exponential trends indicate that the
demand has explosive growth.
Seasonal Influence
Seasonal variation usually occurs within one year and
recurs annually (Schonberger & Knod 1994). Wheelwright and
Makridakis (1985) indicated that seasons may be 3-month
intervals, 30-day intervals, one-week intervals, or even 24-
hour intervals. Seasonal patterns of demand are a good
indicator for making long range forecasts.
19
Cyclical Elements
Cyclical factors are very difficult to predict because
the time span may be unknown or the cause of the cycle may
not be considered (Chase & Aquilano, 1992). Examples of
cyclical factors include political elections, war, economic
conditions, or sociological pressures. Wheelwright and
Makridakis (1985) stated that a cyclical pattern of demand
is similar to a seasonal pattern, but the length of a single
cycle is generally longer than one year. Schonberger and
Knod (1994) explained that a cyclical pattern may be
recurring and often spans several years.
Forecasting Time Periods
Forecasting involves two types of elements: model
development and production demand. Most food service
operations refuse to use quantitative forecast methods due
to misunderstanding the time required in preparing a
forecast or not knowing how to properly use the methods.
Most quantitative forecast models require time, personnel,
and equipment to calculate the output. Some regression
forecast models require a specific period of time to
initiate the model. Developing any forecasting system
requires time and human effort. Repko and Miller's (1990)
study found that very few food service operators used
mathematical models for forecasting demand. Their research
revealed that judgment based on the past records was the
20
most frequently used forecasting method and that production
demand was determined one week in advance. Their study
implied that food service operators are limited to
forecasting methods that are simple and fast.
Long-term Forecasting
A long-term forecast fits neatly into the corporate
strategic planning process. Long-range forecasting
generally predicts two to five years into the future. It is
used in business planning for production, research, capital
planning, plant location and expansion, and advertising
decisions. This type of forecasting is generally broad in
scope and often employs qualitative analysis (Dilworth,
1986).
Intermediate-term Forecasting
The time span for intermediate-term forecasting is
generally in the range of one season to two years. It is
most commonly used in aggregate planning such as in capital
and cash budgets, sales planning, production planning,
production and inventory budgeting. Intermediate-range
forecasting usually uses numerical methods (Dilworth, 1986).
Short-term Forecasting
Short-term forecasting usually predicts future events
for one season, one day, or one year. It is used for short-
21
term control, which includes adjustment of production and
employment levels, purchasing, job scheduling, project
assignment, and overtime decisions. Methods of short-term
forecasting include trend extrapolation, graphical-
projection, personal judgement, and exponential smoothing
(Dilworth, 1986).
Demand Forecasting Models
Types of Forecast Methods
A forecast can range from simply using a "guesstimate"
to complex mathematical methods. Two general categories of
forecast methods are: (1) qualitative and (2) quantitative
approaches. The qualitative approach, sometimes referred to
as the subjective or judgmental method, is based on
subjective assessments. Quantitative, objective, or
mathematic methods include two subgroups of models, time
series and causal models (Chase & Aquilano, 1992; Jarrett,
1991; Wheelwright & Makridakis, 1985).
Oualitative or Subjective Forecasting Methods
The major characteristic of the qualitative approach to
forecasting is human judgment and intuition in an ad hoc
manner. It is sometimes the simplest and fastest way to
forecast. This method involves using only subjective
judgement without expressing the forecast in numerical
terms. Qualitative forecasting methods include the Delphi
22
technique, jury of executive opinion, field sales force,
aggregate subjective forecasts, as well as other methods.
Delphi Technique
Adam and Ebert (1989) stated that "The Delphi technique
is a group process intended to achieve a consensus forecast,
often a technological forecast" (p. 79). Jarrett (1991)
explained that the Delphi method involves using the
subjective opinion of experts to predict the future
direction of economic sectors. This type of technique
avoids direct interpersonal relations and has worked
successfully as a method of technological forecasting.
The Jury of Executive Opinion
The jury of executive opinion approach is one of the
simplest and most widely used forecasting methods
(Wheelwright & Makridakis, 1985; Wilson & Daubek, 1989).
Wheelwright and Makridakis (1985) explained that the jury of
executive opinion approach consists of corporate executives
sitting around a table and deciding as a group what their
best estimate is of future demand. The advantages of this
type of method are that it provides a quick and easy
forecast; it does not require complicated statistics; and it
brings together a variety of specialized opinions. The
drawback of this approach is that since estimators are in
personal contact with another, the weight assigned to each
23
executive's assessment will depend in large part on the role
and personality of that executive in the organization.
Field Sales Force
The field sales force technique requires each sales
representative to estimate the sales within his or her
territory. This method utilizes input from persons in
direct contact with the customer and the field sales force.
This method is most suited for a new product. The advantage
of this approach is that it uses the specialized knowledge
of those closest to the marketplace. The down side for this
method is that it involves individual biases. Often
salespeople are poor estimators and are either overly
optimistic or overly pessimistic (Dilworth, 1986).
Aggregate Subjective Forecasts
The aggregate subjective forecast method is the easiest
and fastest way to estimate demand forecasting. Research
has found that aggregate subjective forecasts are more
accurate than the individual forecast (Ashton & Ashton,
1985; Makridakis & Winkler, 1983). Also, weighing
individual forecasts differentially produces better
aggregate forecasts. Armstrong (1986) stated that expert
opinion is useful in estimating current status; combining
forecasts from extrapolation and judgment methods has been
shown to be highly effective.
24
Other Subjective Methods
Other subjective methods include nominal group
technique, expert opinions, panel consensus, visionary
forecast, and historical analysis. Basically, these
subjective methods are based on a person or a group's
intuition in making predictions rather than the scientific
calculation of future events (Adam & Ebert, 1989; Armstrong,
1986; Makridakis, 1986; Webster, 1986).
Ouantitative or Objective Forecasting Methods
Mathematical forecasting techniques may be effective in
food service operations to control costs, increase
productivity, and maximize profits (Miller, Thompson, &
Orabella, 1991) . Quantitative forecast methods generally
divide into two types: the time series model and the causal
model.
Time Series Models
Time series models assume that patterns reoccur over
time (Wheelwright & Makridakis, 1985). Examples of time
series models include naive, simple average, simple moving
average, weighted moving average, seasonal index, and
exponential smoothing techniques. Studies have found that
very few food service operations utilize quantitative
methods in doing forecasting (Miller, McCahon, & Bloss,
25
1991; Miller & Shanklin, 1988a; Repko & Miller, 1990; Reyna,
Kwong, & Li, 1991).
The Naive method uses the most recent information
available as the actual forecast value. For example, if a
forecast is being prepared for a time horizon of one period,
the most recent actual value would be used as the forecast
for the next period. The formula for a Naive forecast is
simply:
F • = D
Ft+i = The forecast for period t + i,
t = Present period,
i = The number of periods ahead being forecast,
Dt = The latest actual value.
The Naive model assumes that there is no pattern in the
data series to be forecast (Wheelwright & Makridakis, 1985) .
Miller, McCahon, and Miller's (1991) study illustrated that
the Naive model was the least accurate model when applied to
food service forecasting. However, some studies have
indicated that the Naive model based on the judgment of past
data, most recent demand, and intuitive estimate is utilized
by the majority of the food service operators (Miller &
Shanklin, 1988a; Repko & Miller, 1990).
The other type of Naive model considers the possibility
of seasonality in the series (Wheelwright & Makridakis,
1985) . This type of model uses the most recent seasonally
adjusted value as a forecast for the next seasonally
26
adjusted value. The equation for the Naive with seasonality
model of forecast is based on the following:
^j where
Sj = the seasonal adjustment index for season j (or
season j + i).
Miller, McCahon, and Miller (1993) utilized both the
Naive and the Naive with seasonality models in their study
and found that the Naive with seasonality model has a
smaller error than the pure Naive model.
Simple Average. The simple average method is the
average of past data in which the demands of all previous
periods are equally weighted (Adam & Ebert, 1989). The
average demand may be a continuous average or seasonal
average. The example of a continuous average is averaging
past consecutive days demand to estimate future demand
(Bails & Peppers, 1982). An example of a seasonal average
is utilizing the past Monday's average to estimate the
future demand (Messersmith & Miller, 1991). It is
calculated as follows:
Ft = (Di + D2 + ... + Dn) /n
Di = the demand in the most recent period,
D2 = the demand that occurred two periods ago,
Dn = the demand that occurred n-periods ago.
27
Hanke and Reitsch (1989) suggested that simple average
method should be used when the data set has no trend,
seasonality, or other systematic patterns.
Simple Moving Average. A simple moving average
combines the demand data from several of the most recent
periods, their average being the forecast for the next
period (Adam & Ebert, 1989). The simple moving average
method generally takes 4 to 10 past values of a like day of
the week to forecast the future demand. Each week, the
oldest demand is dropped and the most recent is added
(Messersmith & Miller, 1991). The formula for a simple
moving average is simply:
n
^ ^ where
t=I is the oldest period, and
t=n is the most recent period in the n-period average.
Hanke and Reitsch (1989) indicated that the simple
moving average model handles trend and seasonality better
than the simple average model. Chase and Aquilano (1992)
indicated that the simple moving average model is useful in
removing the random fluctuations. They recommended the
simple moving average model for short-term forecasting.
Repko and Miller (1990) found that the moving average is the
most frequently used quantitative forecast method. Simple
28
moving average method is suggested for short-term
forecasting (Miller, McCahon, & Miller, 1993).
Weighted Moving Average. Unlike the simple moving
average method that gives equal weight to each component of
the moving average database, a weighted moving average
allows a weighted constant to be assigned to each element,
so that the sum of all weights equals one (Chase & Aquilano,
1992). This method allows the forecaster to adjust the
effects of past data. Typically, higher weights are
assigned to more recent periods (Schonberger & Knod, 1994) .
The equation for the weighted moving average is as
following:
n
t Z> t t where
E'^t=i t=i
where Ct = Weighted constant, and 0 s C s 1.0.
Seasonal Index. Moving average methods estimate
forecasting by smoothing the past data. The seasonal index
method, however, takes the seasonal factor into
consideration when calculating the forecast (Schonberger &
Knod, 1994) .
Simple Exponential Smoothing. Exponential smoothing
models are easy and often used in operations management
(Adam & Ebert, 1989; Gardner & Dannenbring, 1980) . Simple
29
exponential smoothing models average the past data by
assigning a weighted constant (Gardner & Dannenbring, 1980).
The weighted constant or smoothing coefficient, a, is
between 0 and 1.0 (Schonberger & Knod, 1994). The weighted
scheme applies the greatest weight to the most recently
observed values and lesser weights to the older values. The
formula for the simple exponential smoothing is:
Ft+i = aDt + (l-a)Ft
An alternative way of writing this equation can be:
Ft.i = Ft+ Q?(Dt-FJ .
In this form, the new forecast equals the old forecast
plus a. times the error (Dt - Ft) from the old forecast. If
Of is close to 1, the new forecast will include a substantial
adjustment for any error that occurred in the preceding
forecast. Conversely, when a. is close to 0, the new
forecast will not show much adjustment for a previous
forecast error. Therefore, the effect of a large or small a.
is important to the adjustment of the previous forecast
error (Wheelwright & Makridakis, 1985).
Wheelwright and Makridakis' (1985) study showed that an
01 of 0.1 yields better forecasts than larger values of a.
Also, Schonberger and Knod (1994) suggested that a. should be
in the range of 0.1 to 0.3. Their experiment showed that
exponential smoothing is more accurate than the moving
average method. Makridakis et al.'s (1982) empirical
studies demonstrated that exponential smoothing is quite
30
accurate compared with more complex forecasting methods such
as the Box-Jenkins model.
Adaptive Exponential Smoothing. In adaptive
exponential smoothing, the smoothing coefficient, a, is not
fixed but allowed to fluctuate over time based upon the
pattern of demand changed (Adam & Ebert, 198 9). The
adaptive exponential smoothing model is most effective if it
is computer assisted (Messersmith & Miller, 1991). A
tracking signal is utilized to adjust the value of the a.
It is used to indicate the existence of any positive or
negative bias in the forecast. The cumulative forecast
error is called the running sum of forecast error (RSFE).
The tracking signal is then the RSFE divided by the mean
absolute deviation (MAD) (Schonberger & Knod, 1994):
RSFE Tracking signal =
MAD.
Double Exponential Smoothing (Brown's Exponential
Smoothing). While single exponential smoothing of past data
estimates the forecast, it does not take the trend factor
into calculation. The double exponential smoothing was
introduced by R. G. Brown. This method yields results which
consider the trend observed values (Jarrett, 1991). Jarrett
indicated that Brown's exponential smoothing is more
accurate than either single exponential smoothing or moving
average. The calculation for this type of method would be
most effectively done with computer assistance.
31
Holt's Exponential Smoothing. Similar to Brown's
exponential smoothing. Holt's exponential smoothing is not
just an adjustment to trends but a two-parameter model. A
growth factor is added to the smoothing equation (Jarrett,
1991). Bails and Peppers (1982) stated that these two
parameters must be quantified, although the trial and error
process of finding the best combination of parameters may be
costly and time-consuming. The formula of this model
includes three equations (Hanke & Reitsch, 1989):
1. The Simple Exponential Smoothing formula:
Ft,i= Q?Dt + (l-Q?)Ft.
2. The trend estimate:
Tt.i=i8(Ft.i- Ft) + (l-iS)Tt.
3. Forecast for n periods into the future:
Ft+n= Ft+i + nTt+i
/8= Smoothing constant for trend estimate
Tt+i= Trend estimate
n=periods to be forecasted into future
Ft+n= Forecast for n periods into future.
Studies indicate that Holt's procedure is preferred
over Brown's exponential smoothing (Gardner & Dannenbring,
1980; Jarrett, 1991).
Winter's Exponential Smoothing. While Brown's
exponential smoothing included a trend factor and Holt's
method added a growth factor. Winter's exponential smoothing
is based on the three components of a pattern: randomness/
32
cyclic contrast, linearity, and seasonality (Wheelwright &
Makridakis, 1985). This method employs these three
parameters {a, (3, & 7) through a trial and error approach.
Computer software is very helpful in this type of analysis.
The formulas for this method are the following (Hanke &
Reitsch, 1989):
1. The exponentially smoothed series:
F,=a-^^{l-a) (F,., + r,.,) .
2. The seasonality estimate:
5,=P^M1-P)S,_^.
3. The trend estimate:
T,-y{F,-F,.^)^{l-y)T,_^^
4. Forecast for P periods in the future:
F,^p={F,-PT,)S,
where:
St = seasonality estimate,
St-p = average experience of seasonality estimated, smoothed
to period t-p,
y = smoothing constant for trend estimate,
p= number of seasons.
33
Ft p= forecast for p periods into future.
Causal Models
Causal models find the exact form of the relationship
between independent variables and the dependent variable.
The dependent variable is what the researcher would like to
predict, and the independent variables are the variables
that affect the value of the dependent variable. There are
two types of causal models: linear regression and multiple
regression (Cryer & Miller, 1991; Iman & Conover, 1989).
Armstrong (1986) suggested that causal methods are used only
if historical data are available.
Linear Regression
The linear regression model in forecasting estimates
the nature of the relationship between a dependent variable
and an independent variable. The dependent variable, Y, is
the one to predict, and the independent variable, X, is the
one used to help in the prediction. A simple regression
model can be expressed in the form of a straight line with
the following equation:
Y = l3o + P^X + e
where /So and (3^ are parameters that represent, respectively,
the Y intercept and slope of the regression curve and e is
the random variable between the value of the independent
variable and the regression line (Jarrett, 1991).
34
Multiple Regression
When there is more than one independent variable, such
relationships are called multiple relationships. Hanke and
Reitsch (1989) defined that "Multiple regression is the use
of more than one independent variable to predict a dependent
variable" (p. 200) . The equation for multiple regression is
as follows:
Y= /So + /SA + (3,X, + . . . /8„X, + e
where /8i, (32,...^^ are the regression coefficients explaining
the association between the independent and dependent
variables (Jarrett, 1991). Because of the cost and tedious
labor involved in multiple regression analysis, computer
programs are needed.
Combining Subjective and Objective Forecasting Models
Several studies have found that simple mathematic
methods, such as Naive, simple exponential smoothing and
simple moving average, are as accurate as sophisticated
models, such as double exponential smoothing and simple
linear regression (Armstrong, 1986; Georgoff & Murdick,
1986; Mahmoud, 1984; Miller, McCahon, & Bloss, 1991; Miller,
McCahon, & Miller, 1991; Shahabuddin, 1987; Wheelwright &
Makridakis, 1985) .
It is believed that quantitative methods out-perform
qualitative methods (Carbone & Gorr, 1985; Mahmoud, 1984) .
However some studies found that quantitative methods are not
35
consistently superior in accuracy to judgmental methods
(Lawrence, 1983; Lawrence, Edmundson, & O'Connor, 1986).
Although subjective methods are more widely used by
operations than objective methods, the latter approach is
more accurate than subjective methods (Dalrymple, 1987;
Georgoff & Murdick, 1986). Some researchers claim that
combining forecasts is more desirable than using forecasts
that are prepared by an individual method (Shahabuddin,
1987; Wilson & Allison-Koerber, 1992; Wilson & Daubek,
1989). Because any individual method is difficult to
identify, they add that the accuracy of a combined forecast
depends on which methods and how many are used.
Criteria for Forecasting Models
Cost of Model
According to Spears (1991), the cost of a forecasting
model includes both the development and operational cost.
Development costs relate to constructing the model,
validating the forecast stability, and writing or securing a
computer program. Operational costs include costs incurred
after the model is developed and as it is used. Georgoff
and Murdick (1986) stated that the Naive model is the most
inexpensive to implement and maintain. Moving average and
exponential smoothing techniques require moderate
expenditures. Adaptive smoothing and regression models are
very expensive.
36
The cost of error is another factor to consider in the
selection of a model. Over-forecasting may increase the
food cost and under-forecasting may result in customer
dissatisfaction. The goal of a forecaster is to reach the
optimal region where cost and accuracy can be a trade-off
(Adam & Ebert, 1989) .
Relevancy of Past Data
Spears (1991) stated that the general assumption in
most forecasting is that past behavioral patterns and
relationships will be repeated in the future. In other
words, past data will influence future events only if there
is a clear relationship between the past and future.
Forecasting Lead Time
Forecasting lead time varies according to the type of
operations. Perishable product requires short-term lead
time. Canned goods, however, allow a more flexible lead
time. Lawrence, Edmundson, and O'Connor (1985) found that
the judgmental method is superior to the mathematic method
only if there are long lead times. Studies found that
simple mathematic methods require a short lead time.
(Jarrett, 1991; Mentzer & Cox, 1984; Wheelwright &
Makridakis, 1985; Wilson & Allison-Koerber, 1992).
Regression models are suggested in long-range forecasting
(Mentzer & Cox, 1984; Wilson & Allison-Koerber, 1992).
37
Degree of Stability
The pattern of demand influences the choice of a model.
Different types of operations have different types of
behavioral patterns. Moving average and simple exponential
smoothing methods are best for stable data forecasting
(Georgoff & Murdick, 1986; Miller, McCahon, & Miller, 1993;
Wilson & Allison-Koerber, 1992). A simple moving average
method will perform better than simple exponential smoothing
in forecasting an unstable data pattern (Miller, McCahon, &
Miller, 1993) . Wilson and Allison-Koerber (1992) indicated
that regression models can handle complex patterns.
Availability of Equipment and Facilities
Repko and Miller (1990) indicated that computers are
reliable tools for improving forecast accuracy. Georgoff
and Murdick (1986) stated that computer facilities are not
essential for all qualitative techniques. It is helpful to
have computing facilities for simple mathematic methods. A
computer is essential for adaptive exponential smoothing,
regression, and Box-Jenkins models.
Skills of Personnel
Another consideration in selecting a forecasting model
is the degree of skill required to compute the results.
Wilson and Allison-Koerber's (1992) study indicated that
simple exponential smoothing techniques is less
38
sophisticated than Holt's exponential smoothing, Winter's
exponential smoothing, and linear regression models. The
multiple regression model and Box-Jenkins, however, are the
most complex forecast techniques. They require the
expertise and training of personnel within the organization.
Accuracy
Accuracy is the last and the most important concern in
judging the quality of a forecast. Lawrence (1983)
indicated that there are two main issues concerning forecast
accuracy. The first is whether quantitative techniques are
significantly more accurate than judgmental methods and
secondly, which quantitative techniques are best. An
expensive and sophisticated model is not necessarily more
accurate than a less expensive and simpler model. Multiple
regression is believed to be the best forecast model when it
is used alone (Forst, 1992; West, 1994; Wilson & Allison-
Koerber, 1992; Wilson & Daubek, 1989). Studies found that
forecast accuracy improved when more methods are involved
(Armstrong, 1986; Lawrence, Edmundson, & O'Connor, 1986;
Makridakis, 1981; Reyna, Kwong, & Li, 1991; West, 1994;
Wilson & Allison-Koerber, 1992) . However, too many methods
may mean confusion. How to select the right methods and
make good combinations is a challenge to the decision
support system (West, 1994).
39
Measuring the Accuracy of Forecast Models
The goal of a forecaster is to minimize the forecast
error. Thus, the error or deviation is defined as:
Error = actual - forecast
or
Et = Dt - Ft
Et = Error for period t (Schonberger & Knod, 1994).
Some of the most common indicators of accuracy are the
bias, mean absolute deviation (MAD), mean square error
(MSE), root mean square error (RMSE), and mean absolute
percentage error (MAPE) (Adam & Ebert, 198 9).
Bias
One of the methods to measure error is called bias,
which is the average of errors, and is given in the
following equation:
n
Bias=^^ j : (F,-D,)
n
where
Ft = Forecast for period t,
Dt = Actual demand that occurred in period t.
Bias indicates the directional tendency of forecast
errors. For example, if a forecast has been overestimating
constantly, it will have a positive value of bias.
40
Consistent underestimation will produce a negative value of
bias (Adam & Ebert, 1986).
Mean Absolute Deviation (MAD)
Unlike bias, the Mean Absolute Deviation (MAD) provides
an accurate measure of the magnitude of forecast error. MAD
disregards the plus or minus sign and measures errors
between forecast and mean demand. The formula is given as
the following (Schonberger & Knod, 1994):
EN^.I MAD=-^^^'
n
where
Et = Error for period t
Mean Square Error (MSE)
Similar to MAD, MSE is also an average of forecast
error, but because its squaring function, the effect of the
direction of the difference is removed (Miller, McCahon, &
Miller, 1991). This measure defines error as the sum of the
squares of the forecast errors divided by the sample size.
The equation is as the following (Schonberger & Knod, 1994):
MSE^^^ ^ E _
n
where
41
F = Forecast for period t,
D = Actual demand that occurred in period t.
MSE is one generally accepted technique for evaluating
exponential smoothing techniques (Jarrett, 1991).
Root Mean Square Error (RMSE)
Another method for calculating forecast error is called
root mean square error (RMSE), which is the square root of
MSE. This method measures error in terms of units which are
equal to the original values studied (Jarrett, 1991). The
formula is as follows:
RMSE-'\
E ( .- .) n
where
F = Forecast for period t,
D = Actual demand that occurred in period t.
Mean Absolute Percent Error (MAPE)
Another accuracy measure is the mean absolute percent
error (MAPE), which combines the individual percentage
errors without offsetting the negative and positive values.
The assumption for this measurement is that the severity of
error is linearly related to its size. It is defined by
Schonberger and Knod (1994) as:
42
" \E I E (--) xl00%
MAPE=-^ n
Et = Error for period t,
Dt = Actual demand that occurred in period t.
Schonberger and Knod (1994) indicated that MAPE uses a
common percentage format and it is good for comparing
forecasting models. However, Jarrett (1991) states that
MAPE is less valid than either MSE or RMSE except for the
simplifying assumption, that the severity of error is
linearly related to its size. Therefore, this measure is
appropriate whenever the error is linear and symmetric.
Regardless of the measure being used, the lowest value
generated indicates the most accurate forecasting model.
CHAPTER III
METHODOLOGY
The review of literature supports a need for a
mathematical forecast model in food service operations,
although, few operations utilize such a forecast technique.
The purpose of this study was to compare the various
forecasting methods and then determine which was the most
appropriate method for the SFSP in west Texas. To screen
the data set, a time plot of each site was produced. Trend
and seasonality of the demand pattern of each site was
studied prior to applying various forecasting techniques.
Pilot Study
Prior to this study, a pilot study was conducted to
analyze data from one of CEI's cafeterias located in Vernon,
Texas. The criteria used in selecting the best forecast
model for CEI were the pattern of demand, accuracy, and
simplicity of the forecast model. The pilot study used Mean
Absolute Deviation (MAD) and Mean Absolute Percent Error
(MAPE) in measuring the accuracy of the forecast model.
In the pilot study, the researchers utilized data
collected during the summer of 1994. The data was provided
on the program's food production record (Appendix A). The
researchers highlighted the number of total meals served
including children, program adults, non-program adults, and
43
44
menu item from the food production record for breakfast and
lunch and entered these numbers into a spreadsheet. The
coding of the data is summarized in Appendix B. The site
was closed on July 4 due to the national holiday. The
researchers used the average demand for Mondays as the
missing value. The daily demand for meals was expressed in
the form of a time plot. This screening provided the
researchers a picture of the trend demand of the operation.
Weekly and days of week demand patterns were extracted from
the data. The servings for each menu item also were
calculated.
The literature (Chase & Aquilano, 1992; Hanke &
Reitsch, 1989; Iman & Conover, 1989) suggested that simple
time series models outperform more complex forecast methods.
Also, Schonberger and Knod (1994) recommended that moving
average with n=5 produced a better forecast than n=3.
Several forecasting methods (moving average with n=3, n=5,
simple exponential smoothing with a=0.2, and Winter's
exponential smoothing with optimized a, /8, & y) were applied
in the analysis of the data. MAD and MAPE were calculated
for each forecast method to compare the accuracy of the
various methods. MAD and MAPE were calculated by using the
spreadsheet from Microsoft Excel (1993) version 5.0. One of
the important findings from this pilot study was that the
moving average method with n=5 produces a better forecasting
result than a moving average method with n=3.
45
After completing the pilot study, some changes were
made for the present study. Weekly demand data were not
needed because the daily demand pattern provided the
information needed for screening the pattern of the demand.
The method for filling the missing value for July 4 was re
evaluated. When the pattern of the demand is unknown, it is
inappropriate to average the days of demand for the missing
values because the future demand may be different from the
past data (Bremer, personal communication, May 22-28, 1995) .
Bremer (1995) suggested using the simple exponential
smoothing method to fill in the missing values. Also, in
the pilot study, the researchers used a=0.3 as the smoothing
constant for the simple exponential smoothing method.
Further study was recommended to ascertain the optimized
constants for all exponential smoothing methods in order to
minimize forecast errors.
In measuring the accuracy of a model, the researchers
used MAD and MAPE. Mean Square Error (MSE) has been
commonly utilized as an accuracy measurement (Jarrett, 1991;
Miller, McCahon, & Bloss, 1991; Miller, McCahon & Miller,
1993; Miller, McCahon, & Miller, 1991). Root Mean Square
Error (RMSE) is the square root of the MSE, which measures
the equivalent units of observation of MAD and MAPE;
therefore, the researcher incorporated RMSE in the present
study to provide another objective accuracy measurement.
The time plot from the pilot study showed that the trend
46
decreased toward the end of the summer. Jarrett (1991)
indicated that Holt's exponential smoothing incorporates the
trend factor in forecasting. Therefore, Holt's exponential
smoothing was incorporated into the present study.
Sites
The research analyzed data from all cafeterias
throughout west Texas operated by CEI. Seven cafeterias
were in operation in 1992 and 10 for the summers of 1993 and
1994. Sites in the two largest communities rotated between
several locations. In Midland, Texas, sites operated each
summer in one of two elementary schools. In Odessa, Texas,
sites rotated among three of the four Boys Club locations.
As a result, only three of the sites opened in 1992 were
also opened during the summers of 1993 and 1994. Each
cafeteria served from 50 to 350 children per meal.
Data Collection
Data for this research study were collected during the
summer of 1992 through the summer of 1994. The data were
collected by CEI staff on the day of each meal service
including breakfast and lunch. The data were provided to
the researcher by the director of CEI.
47
Treatment of Data
Data from meal production records from 1992 through
1994 were highlighted and entered into a spreadsheet. None
of the data provided were modified or weighted. Substitutes
for the missing values for July 4 of each summer was
produced by using the simple exponential smoothing method
with a= 0.3. Sites, days of week, weeks of operation, and
menu items were coded for statistical purposes. The coding
of the data is summarized in Appendix B.
Data Analysis
Phase I--Selection of an Appropriate Forecast Model
After data collection was completed and then compiled
into a spreadsheet, the data was analyzed based on the three
criteria specified for the model. The review of literature
revealed that the pattern of the past data is crucial to the
selection of a forecast model. Accuracy, of course, is the
overall goal of forecasting. The nature of SFSP in the west
Texas demands a minimum of training and resources for the
implementation and administration of a forecast model.
Therefore, simplicity factor was added as a criteria in
selecting forecasting method of this study.
With the previous considerations, three criteria were
established in selecting an appropriate forecast method:
pattern of the time plot, accuracy, and simplicity. A scale
of 1-4 (l=lowest ranking and 4=highest ranking) was used to
48
evaluate each forecasting method for each of the three
criteria. Also, each criterion was weighted as follows:
30% for pattern, 30% for accuracy, and 40% for simplicity.
All sites were analyzed separately. Based on the selection
from each individual site, a final recommendation of a
forecast method for breakfast or lunch followed the site
analysis. The most frequently selected method from
individual sites would then be recommended to the SFSP in
west Texas.
First, the analysis included a study of the pattern of
the time plot (Appendix D) for each site for each meal
served. By using the Execustat (1993) statistic program, a
separate time plot for breakfast and lunch was produced for
each site. This preliminary screening provided an overview
of the pattern of the time plot at each site. The time plot
for each site was examined for trend or seasonality
patterns. These individual time plots were examined by
visually analyzing the pattern of the data set.
The preliminary screening provided an assessment of the
strengths of any particular model. By visualizing the time
plot, when the time plot showed a decreasing or increasing
trend, the value for trend was assigned 1 (assigned trend
and seasonality values were included in Appendix D). On the
other hand, if there was no trend, the value for the trend
was 0. Seasonality of the time plot was defined as the
ratio of one seasonal index from an individual time plot to
49
the whole population's seasonal index distribution. The
seasonal index was defined by an average day's demand
divided by the weekly demand. For example, if a site's
lowest seasonal index is .90, and the highest index is 1.18,
the seasonal index for this site fluctuated from the lowest
point to the highest point 28%. This fluctuation percentage
is recorded and divided by the highest fluctuation for all
sites, e.g. 38%. The seasonal value, in this instance, was
28% and was divided by 38%, resulting in 73.68%. At this
point, a seasonal value for all sites were produced. The
seasonal value of individual site indicated the strength of
seasonality. The researcher used 50% as the cut off to
determine the seasonality of pattern. If the seasonal value
was higher than 50%, the value for seasonality for this
particular time plot was assigned 1. On the other hand, if
its seasonal value was below 50%, then the value would be 0.
Winter's method adapts trend and seasonality better
than the Holt, SES, or MA methods. The Holt method,
however, detects the trend factor better than SES and MA
(Adam & Ebert, 1989; Georgoff & Murdick, 1986; Hanke &
Reitsch, 1989; Jarrett, 1991; Wheelwright & Makridakis,
1985; Wilson & Allison-Koerber, 1992). Therefore, if the
time plot showed a trend pattern but no seasonality from the
preliminary sorting, the Winter's method got 3 points.
Holt's got 4 points, SES got 2 points, and MA got 1 point.
On the other hand, if the pattern showed no trend and no
50
seasonality, MA received 4 points, SES received 4 points.
Holt's received 2 points, and Winter's received 1 points
(Hanke & Reitsch, 1989; Bremer, personal communication, June
10, 1995) . The higher the score, the greater weight a
method had for pattern of time plot. Table 1 summarizes the
ranking criteria for pattern of the time plot.
The second step in the analysis was to compare the
accuracy of the various forecast models. The experimental
forecasting models included moving average (MA) (with n=5),
simple exponential smoothing (SES), Holt's exponential
smoothing (Holt), and Winter's exponential smoothing
(Winter). Formulas for these forecasting methods were
presented in Chapter II. MAD, MAPE, and RMSE were utilized
to measure the accuracy of a forecast model. Formulas for
calculating these measurement tools also were provided in
Chapter II. Data were analyzed with a Microsoft Excel
(1993) spreadsheet program or Execustat (1993) statistic
program based on the calculation that was appropriate for
each model. Optimized smoothing constants {a, /8, & y) were
generated by the Execustat (1993) statistic program to
ensure the best results of MAD, MAPE, and RMSE from
exponential smoothing methods. The forecast, forecast
error, MAD, MAPE, and RMSE were produced along the same
spreadsheet or the Execustat (1993) statistic program.
51
Table 1 Ranking values* of forecast methods based on objective and subjective analysis for pattern of time plot.
Method
Pattern MA SES Holt Winter
Trend & Seasonality
Trend Only
Season Only
No Trend & No Seasonality
1
1
2
4
1
1
2
4
3
4
2
2
4
3
4
1
Scale l=lowest ranking and 4=highest ranking
52
Accuracy for each site was determined based on the
lowest values for MAD, MAPE, and RMSE. However, for an
individual site, the results based on MAD, MAPE, and RMSE
might differ. Therefore, a preliminary ranking value was
assigned to the MAD, MAPE, and RMSE results. The final
ranking for accuracy for each site was based on the sum of
the rankings from MAD, MAPE, and RMSE; the highest sum
received 4 points, and the lowest sum received 1 point.
Table 2 illustrates the results of ranking the methods for
accuracy for an individual site.
A value for simplicity was assigned to each method. MA
and SES have the same high level of simplicity; therefore,
they both received 4 points. Because Winter's method is the
most complicated, 1 point was assigned to this method. Two
points were assigned to Holt's method (Adam & Ebert, 1989;
Bremer, personal communication, June 10, 1995; Georgoff &
Murdick, 1986; Hanke & Reitsch, 1989; Jarrett, 1991;
Wheelwright & Makridakis, 1985; Wilson & Allison-Koerber,
1992) .
For each site, a comparison of all models based on
values assigned for the pattern of time plot, accuracy, and
simplicity was produced. Weighted comparisons were made
with a emphasis on simplicity of the model. The selection
of a forecast model for each site was made based on these
comparisons with the highest total score determining the
method selected. Table 3 illustrates the final selection of
53
Table 2. Example for one site: Ranking values* for accuracy of forecast methods.
Method
MA
SES
Holt
Winter
MAD
7.27
6.99
7.14
7.44
Ri
2
4
3
1
MAPE
6.05
5.75
5.68
5.83
R2
1
3
4
2
RMSE
20.21
19.48
19.23
20.24
R-1
2
3
4
1
^K
5
10
11
4
Ea
2
3
4
1
Scale l=lowest ranking and 4=highest ranking
Table 3 Example for one site: Selection of an appropriate forecast method based on pattern, accuracy, and simplicity.
Method
MA
SES
Holt
Winter
Pattern
2
2
2
4
Accuracy
2
3
4
1
Simplici
4
4
2
1
ty Total*
2.8
3.1
2.6
1.9
* Pattern * 0.3 + Accuracy * 0.3 + Simplicity * 0.4
54
the most appropriate forecast method for an individual site.
Since the factors utilized in the data analysis were
given specific weights, the resulting scores allowed an
objective, quantitative assessment of each of the forecast
methods. The final selection of the forecast method for all
sites was based on the highest mean ranking score of the
methods. The final product of this phase may not
necessarily be the best method for a particular site;
however, the aggregate forecasting model will be the one
most appropriate overall. This final product will be the
one recommended to the SFSP in west Texas to estimate
forecasting.
Phase II--Evaluation of Menu and Daily Indices on the Accuracy of a Forecast Model
The second phase of the study analyzed the effect of
menu preference and day of demand in influencing the
forecast. Three sites. Central, DeZavala, and Tulia, were
chosen for this analysis. These three sites were selected
because data were consistently collected at these sites
during all three summer periods reviewed.
The first step of this phase was to calculate the daily
index and the menu index for each site. The daily index was
defined on Phase I of this study as the seasonal index. The
researcher averaged the daily indices from 1992 and 1993 for
these three sites. The averaged indices were then applied
to the data collected in 1994 along with the best forecast
55
model selected from the Phase I of this study. The equation
utilized for determining the most accurate forecast is
expressed as followings:
Forecast(„ew) = Forecast( id) * Index.
Forecast (new) was defined as the forecast value after
applying the index value, and Forecast( id) was defined as the
best method from the Phase I evaluation. A comparison of
forecasts with and without these daily indices was then
produced. The MAD, MAPE, and RMSE were applied to determine
if applying these indices improves the forecasting accuracy.
The menu index was defined as the actual demand divided
by the estimated forecast value from the forecast method
being selected from Phase I (Iman & Conover, 1989).
Breakfast and lunch menu indices were analyzed separately.
If the menu index was greater than 1, under-forecasting
occurred. Therefore, the estimated forecast needed be
adjusted by this index. On the other hand, if the menu
index was less than 1, which indicated an over-forecasting,
the estimated forecast needed be adjusted by the index. If
the menu index was consistently high, the estimated forecast
was then adjusted by the index. To evaluate the
effectiveness of the menu index, data from Central,
DeZavala, and Tulia of 1994 was analyzed using the forecast
method selected in Phase I. Estimated forecasts were
56
adjusted by these indices. MAD, MAPE, and RMSE were
calculated before and after indices adjustments were
applied.
If the two indices, daily and menu, showed a
significant contribution to the accuracy of the forecast
from Phase II of this study, the estimated forecasts should
be adjusted by the menu and daily indices. This would
result in a more accurate forecast.
CHAPTER IV
RESULTS AND DISCUSSION
The purpose of this study was to identify an
appropriate forecast method for the Summer Food Service
Program (SFSP) in west Texas. This study compared four
forecast methods, moving average (MA), simple exponential
smoothing (SES), Holt's exponential smoothing (Holt), and
Winter's exponential smoothing (Winter). The criteria for
selecting the most appropriate method included the pattern
of time plots, accuracy of forecast models, and simplicity
of models. After the most appropriate forecast method was
determined, daily and menu index adjustments were made to
result in the optimal forecasting. The data for this study
were collected over the summer of 1992 through the summer of
1994.
Pattern of Time Plots
The review of literature revealed that the pattern of
time plots is very important in selecting a forecast method.
Trend and seasonality were two factors that determine the
pattern of a time plot. Tables 4 and 5 illustrate the
frequency of trend or seasonality versus no trend or
seasonality from all SFSP sites. In the analysis of sites
that manifest trends versus the absence of trends. Table 4
shows a bias in favor of no trend when breakfast and
57
58
Table 4.
Meal
Breakfast'
Lunch**
Total
Trend analysis SFSP sties.
1
Number of Sites
11
13
24
of time
Trend
Percent
46%
48%
47%
pl ots for meals served at
No Trend
Number of Sites
13
14
27
Percent
54%
52%
53%
24 sites for breakfast
^ 27 sites for lunch
Table 5. Seasonality analysis of time plots for meals served at SFSP sites.
Meal
Breakfast*
Lunch^
Total
Seasonality
Number of Sites
10
14
24
Percent
42%
52%
47%
No
Number of Sites
14
13
27
Seasonality
Percent
58%
48%
53%
^ 24 sites for breakfast
^ 27 sites for lunch
59
lunch are combined. However, the total difference between
sites showing trends and those that do not is only 6%. For
breakfast, the sites that show no seasonality (58%) are
greater in number than those that do (42%) . For lunch, the
distribution between the two values is about even. Again,
there is a bias in favor of no seasonality in combining
breakfast and lunch, but the discrepancy is minor (only 6%
difference). This minor discrepancy of various patterns
indicates that the pattern of the SFSP in west Texas is not
consistent.
Accuracy of Forecast Models
Root Mean Squared Error (RMSE) , Mean Absolute Deviation
(MAD), and Mean Absolute Percentage Error (MAPE) were used
in measuring the accuracy of forecast methods. For each of
these techniques, the smaller the forecast error, the more
accurate the forecast model. Tables 6, 7, and 8 show the
comparison of accuracy based on RMSE, MAD, and MAPE,
respectively, for breakfast and lunch. In the analysis of
both breakfast and lunch. Table 6 shows that SES has the
lowest forecast errors. Also, Table 7 illustrates that SES
method has the lowest forecast errors among the forecast
methods. Table 8, however, shows that when using MAPE as
the measure of error. Holt method has the lowest forecast
errors. Therefore, simple exponential smoothing
Table 6. Analysis of RMSE by forecast method for meals served at SFSP sites.
60
Forecast Method
MA
SES
Holt
Winter
Breakfast*
Mean of RMSE
10.90
10.07=
10.33
12.38
i
SD
6.06
5.37
5.37
6.99
Lunch^
Mean of RMSE
19.87
18.63=
18.68
21.52
SD
9.69
9.41
9.31
10.54
* 24 sites for breakfast
^ 27 sites for lunch
= Indicates the most accurate model
Table 7. Analysis of MAD by forecast method for meals served at SFSP sites.
61
Forecast Method
MA
SES
Holt
Winter
Breakfast*
Mean of MAD
7.77
7.23=
7.56
8.99
SD
3.91
3.32
3.75
4.85
Lunch**
Mean of MAD
14.88
14.15=
14.36
16.68
SD
7.26
7.31
7.05
8.40
* 24 sites for breakfast
^ 27 sites for lunch
= Indicates the most accurate model
62
Table 8. Analysis of MAPE by forecast method for meals served at SFSP sites.
Forecast Method
MA
SES
Holt
Winter
Breakfast^
Mean of MAPE
27.43
23.14
23.06=
25.80
I
SD
21.69
15.01
14.07
14.84
Lunch^
Mean of MAPE
12.63
11.98
11.89=
13.48
>
SD
4.54
4.27
4.07
4.01
24 sites for breakfast
^ 27 sites for lunch
= Indicates the most accurate model
63
outperformed the other methods for forecast accuracy based
on two of the tree measures utilized. Table 8 shows that
the mean MAPE for breakfast is higher than the mean MAPE for
lunch. This was due to the variability of breakfast data.
The average number of serving for breakfast is about 45.
Therefore, 25% of error will contribute about 10 servings of
error. The purpose of this study was to provide a forecast
tool for foodservice operators to utilize. Human judgment
might be required to implement a more accurate forecasting.
The Most Appropriate Forecast Method
After examining the pattern of time plots, determining
the accuracy of forecast models, and rating the simplicity
of the methods, the final selection of an appropriate
forecast method was made. To make the selection, the
procedure illustrated on Table 3 was applied for each site,
and the final scores were computed by averaging all sites'
total scores for each method. Figure 1 and Figure 2 provide
a visualized comparison of the forecast methods for
breakfast and lunch respectively. The box contains the
majority mean score distribution (McClave & Benson, 1994).
The majority mean score distribution, illustrated by the
box, shows that the SES method has the highest score among
methods for both breakfast and lunch. The plus signs,
indicating the mean score for each method (McClave & Benson,
64
Breakfast S
core
4
3.5
3
2.5
2
1.5
1
.,
+
+ n
- T -
^ • -*-
-
• -
-
+ J
-
f - i LJ
1 L 2 Method
3 it
4
Figure 1. Final selection of forecast method for breakfast.
* Method: 1=MA; 2=SES; 3=Holt; 4=Winter + Mean score • Indicates an outlier
65
u o u CO
3.5
3 -
2.5 -
1.5 -
1 -
Lunch
--
-1
•
•
-
-
' •
+
+
.
. • -
D
-
+
-
D
-
-
-
+
-
-
-
-
-
2 3
Method*
Figure 2. Final selection of forecast method for lunch.
* Method: 1= MA; 2=SES; 3=Holt; 4=Winter + Mean score D Indicates an outlier
66
1994), illustrates that SES has the highest mean score for
both breakfast and lunch. In summary, SES has the highest
mean score among the methods for both breakfast and lunch.
Table 9 provides a summary of the ranking of the resulting
scores. From Table 9, it is clear that simple exponential
smoothing had the highest mean ranking scores indicating it
is the most appropriate forecast method for the SFSP.
Alpha Analysis
The SES forecasting method requires the utilization of
Of as a smoothing constant. Since SES was selected during
Phase I of the study as the most appropriate forecast method
for the SFSP, an analysis of alpha value was conducted.
Figure 3 shows the distribution of optimal a values for all
sites when SES was utilized. The mean a for breakfast is
close to 0.40 and the mean a for lunch is close to 0.50.
Further analysis was conducted to ascertain the best o? for
sites having trends versus those that do not. Table 10
displays the analysis of a for both breakfast and lunch.
The value of a=0.50 has the greatest frequency in various
time plot patterns. Table 10 suggests that if one site had
no trend pattern for breakfast in the past, a = 0.35 should
be used; if its past pattern showed a trend, a = 0.52 should
be used. If the site has seasonality in the past for
breakfast demand pattern, a = 0.34 should be used. For
those sites with no seasonality for breakfast in the past.
67
Table 9 Comparison of mean ranking score for appropriateness of the forecast methods for breakfast and lunch.
Forecast Method
MA
SES
Holt
Winter
Breakfast*
Mean Ranking Score=
2.86
3.24**
2.52
1.71
SD
0.42
0.42
0.31
0.41
Lunch*'
Mean Ranking Score=
2.90
3.28"
2.43
1.76
SD
0.49
0.53
0.38
0.56
* 24 sites for breakfast
** 27 sites for lunch
= Pattern * 0.3 + Accuracy * 0.3 + Simplicity * 0.4
• Indicates the most appropriate model
68
Breakfast
0.4 -I I 1 — I I I — I — I I I — I I — I — I — I — I — I »
0.2
u u u 0
0.2
0.4 I • • • t I — I 1 — I — I 1 — I — I 1 — I — L
0 0.2 0.4 0.6 0.8 1 Lunch
Figure 3. Alpha distribution for simple exponential smoothing method.
69
Table 10. Comparison of mean alpha for different patterns of time plot for breakfast and lunch.
Breakfast Lunch
Pattern Mean Alpha
0 .
0 .
0 .
0 .
52
35
34
50
Mean Alpha
Trend
No Trend
Seasonality
No Seasonality
0.54
0.50
0.52
0.53
70
Q; = 0.50 should be used. If it is a new site, since
management cannot yet determine a pattern for future demand,
a = 0.50 will be appropriate.
In summary, sites for breakfast having trend patterns
should have higher values of a than those without trends
when applying SES. However, sites with seasonality should
have lower a values than sites with no seasonality in
breakfast. For lunch service, the mean a is approximately
equal to 0.5 for all patterns.
Daily Index
Phase II of this study examined the impact of the days
of demand and the menu item on the accuracy of the forecast
method. After the appropriate forecast method, SES, was
selected in Phase I, the daily index was then calculated and
applied to the SES forecast. When applying simple
exponential smoothing with a = 0.40 for breakfast and a =
0.50 for lunch to the data from Central, DeZavala, and
Tulia, averaged daily indices from 1992 and 1993 were
utilized to adjust the forecast for 1994. Comparisons of
the various computations of forecast error (RMSE, MAD, and
MAPE) with and without adjustment of the indices are given
on Table 11 and Table 12. Positive entries in the %
Improved column in Tables 11 and 12 indicate an improvement
in accuracy with the daily index; negative entries indicate
the reverse. The results illustrated in Tables 11 and 12
71
Table 11 Forecast error measures with adjustment of daily index for breakfast.
Sites RMSE Adjusted RMSE
o Improved
Central DeZavala Tulia
4.00 12.93 N/A
4.00 12.90 N/A
0.0% 0.2% N/A
MAD Adjusted MAD
o. Improved
Central DeZavala Tulia
3.22 9.11 N/A
3.29 9.10 N/A
-2.2% 0.1% N/A
MAPE Adjusted MAPE
o, o Improved
Central DeZavala Tulia
33.05 42.50 N/A
32.89 42.06 N/A
0.5% 1.0% N/A
72
Table 12 Forecast error measures with adjustment of daily index for lunch.
Sites RMSE Adjusted RMSE
Improved
Central DeZavala Tulia
7.31 31.76 18.11
7 31 18
24 16 67
1.0% 1.9%
-3.1%
MAD Adjusted MAD
% Improved
Central DeZavala Tulia
6 26 13
11 33 07
5.97 25.66 13.85
2 . 2 *? 2.5% -6.0%
MAPE Adjusted MAPE
g. Improved
Central DeZavala Tulia
17 11 10
09 19 54
16.88 11.09 11.08
1.2% 0.9 g. -5.1%
73
indicate that there was no significant improvement in
accuracy when these daily indices were applied. Table 12
shows that the Tulia site even had a negative value for %
Improved when the daily indices were applied.
Menu Index
A menu index is used to adjust a forecast based on the
popularity of actual menu items offered. The forecasted
values for menu items should be adjusted by the index if the
menu index deviates from 1 (index=l indicates a perfect
forecast) . The box plots in Figure 4 contain 50% of the
observations for each menu item. Figure 4 also shows that
most mean scores are between 0.9 and 1.1 (the plus sign
inside the box plots indicates the mean of the menu index) .
Based on Iman and Conover (1989) , if the menu index was
consistently above 1, under-forecasting occurred. On the
other hand, if the menu index was consistently below 1,
over-forecasting occurred. When the menu items were bagels
and cereal (breakfast menus item 1 and 2) the indices were
consistently below the average index, 1. The mean of menu
indices for toast (breakfast menu item 4) was below 1;
however, the median (the bar line inside of each box plot)
of the menu index was above 1 (Figure 4) . The median of
menu index above 1 means that half of the sites that served
toast have been over-forecasting but on average, under-
forecasting has occurred. Breakfast pizza (breakfast menu
74
I
1.6 -
1.4 -
1.2 -
0.8 -
0.6 -
0.4 -
1 - -
1 2 3 4 5 6 7 8 9 10 11 12 Breakfast Menu*
Figure 4. Menu index analysis of breakfast
* Code of breakfast menu item on Table 13 ** Mean of forecasting effectiveness + Mean score D Indicates an outlier * Indicates an extreme outlier
75
item 12), has the highest index when compared to other menu
items. The other menu items showed no consistently high or
low indices. Table 13 summarizes the mean of menu index.
The results of this analysis (Table 13) indicate that when
the menu items were bagels or cereal, the indices were 6%
below the average; breakfast pizza, however, was 8% above
the average index. Therefore, if the menu item is breakfast
pizza, the forecast should be adjusted by 1.08.
The lunch menu index (Figure 5) shows a pattern similar
to the breakfast index. The majority of the menu indices
are located between 0.9 and 1.1. Pizza (lunch menu item 6)
has a consistently higher index than the other menu items.
The corn dog, spaghetti, and chicken salad sandwich items
(lunch menu items 3, 7 and 21) were under-forecasted. As
illustrated in Table 14, the lunch menu index fluctuated
between 0.96 and 1.07.
From the analysis described above, there were no
consistently popular or unpopular menu items for either
breakfast or lunch. Since there were only 6 sites offering
breakfast pizza or lunch pizza, the observations were
insufficient to determine solid recommendations for
adjusting the forecast. Ideally, when determining the index
for menu items, the same number of observations is available
for all menu items. Because the number of times that menu
items were served during the study period fluctuated, the
validity of the menu indices may have been effected. Bremer
76
Table 13. Breakfast menu index analysis for selected sites of SFSP.
Menu Item Number* Index
1. Bagel 15 0.94
2. Cereal 24 0.94
3. Donut 24 0.98
4. Toast 14 0.96
5. Cinnamon roll 15 0.94
6. Grits 4 0.92
7. Oatmeal 12 0.95
8. Muffin 10 1.00
9. Pancake 7 0.98
10. Eggs 6 1.00
11. Bacon w/break 2 1.29
12. Breakfast Pizza 6 1.08
Number of times the menu item was served
77
I ITS
1.4 -
1.2 -
1 -
0.8 -
0.6 -
1 3 5 7 9 11 13 15 17 19 21 2 4 6 8 10 12 14 16 18 20
Lunch Menu*
Figure 5. Menu index analysis of lunch
* code of lunch menu items on Table 14 ** Mean of forecasting effectiveness + Mean score D Indicates an outlier * Indicates an extreme outlier
78
Table 14. Lunch menu index SFSP.
Menu
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
Item
BBQ
Chicken nugget
Corn Dog
Hamburger
Hot Dog
Pizza
Spaghetti
Taco
Ham & Cheese sandwich
Cheese sandwich
Turkey sandwich
Peanut butter & Jelly
Roast beef sandwich
Tuna sandwich
Burrito
Steak finger
Macaroni & Cheese
analysis for
sandwich
Bologna & Soloni sandwich
Fish nugget
Chicken sandwich
Chicken salad sandwich
selected
Number*
21
20
20
20
1
20
20
20
11
12
7
12
2
8
12
12
3
6
8
9
6
sites of
Index
0.97
1.02
0.96
1.00
0.98
1.05
0.96
1.00
1.01
1.01
1.02
1.00
1.07
0.98
1.01
1.01
1.02
0.98
0.98
1.01
0.97
* Number of times the menu item was served
79
(1995) suggested that because these were only a few items
that were over-forecast or under-forecast, it was determined
that incorporation of the menu item index into the forecast
model was not necessary.
Recommendation to SFSP in West Texas
Based on the three original criteria for the forecast
method for the SFSP, the SES method with a = 0.50 was
recommended. Appendix C illustrates a worksheet that was
developed so that SES could be implemented by the SFSP. In
order to minimize training requirements, this worksheet was
designed for SES with a = 0.50 and can be used to estimate
either breakfast or lunch demand.
Comparison of Results
The results of this study shows that SES and MA, simple
mathematic forecast models, outperformed the more complex
models of Holt's and Winter's methods. Schonberger and Knod
(1994) suggested that not only are the simple mathematic
forecast methods easier to implement but in some cases may
be more accurate than the complicated methods. Thus, the
results of the present study and Schonberger and Knod (1994)
are in agreement about the relationship of accuracy and
complexity of forecast models.
Miller, McCahon, and Bloss (1991) conducted research in
a university dining hall and employed mean square error
80
(MSE) and mean absolute deviation (MAD) as the accuracy
measurements. Their research indicated that the simple
moving average (MA) forecast method is less accurate than
simple exponential smoothing (SES) method. The results of
the present study agree with Miller, McCahon, and Bloss's
(1991) study that SES is more accurate than the MA method
(Table 6, 7 and 8).
Miller, McCahon, and Miller (1991) conducted research
in a commercial foodservice environment to test the accuracy
of several simple mathematical forecasting models. MSE and
MAD was utilized in their study to measure the forecast
accuracy. Their study indicated that SES is more accurate
than the double exponential smoothing model, which is
similar to Holt's model. Similarly, the results of the
present study showed that Holt's method was less accurate
than SES method when testing with RMSE and MAD (Table 6 and
Table 7).
In a study to compare long-term and short-term
forecasting, Miller, McCahon, and Miller (1993) suggested
that a small a is better for stable data. Also, Schonberger
and Knod (1994) recommended an a in the 0.1 to 0.3 range for
stable demand patterns. However, in the present study, the
analysis of a value for SES was found to be 0.50. The
analysis of alpha value of the present study suggested that
if the operation had no trend or no certain a future trend,
a small a may be appropriate. Gardner (1980) noted that
81
higher values of a, between 0.5 and 0.75, detected the
development of trends faster than a smaller a. As
illustrated in Table 4, the trend distribution among all
sites is half and half. Therefore, the o; value equals 0.5
may be the best constant for applying SES for this
application.
CHAPTER V
CONCLUSIONS
Forecasting is an important element of a foodservice
operation. This study identified a simple and accurate
method appropriate for implementation by a SFSP. Prior to
this research, there was no forecasting method utilized by
the SFSP, and over-forecasting occurred often. Waste of
resources, including labor and food, resulted in a waste of
federal funding and, ultimately a waste of taxpayers' money.
A major reason for not using a mathematic method in
forecasting is the fear of complexity. The results of this
study showed that effective forecasting need not require
complex calculations. The results of this study may not
only apply to the SFSP but may also be applicable for small
and short time span food service operations as well. This
research design and methodology also could apply to other
types of hospitality operations.
Major Findings of the Study
An interesting finding from this research was that for
this application, simple mathematic forecast methods, such
as simple exponential smoothing (SES) and moving average
(MA), are at least as accurate as more sophisticated methods
such as Holt's exponential smoothing (Holt) and Winter's
exponential smoothing (Winter) methods. The SES was not
82
83
only the most appropriate forecast method for the SFSP but
also was the most accurate forecast method.
The best a for breakfast was shown to be 0.4 and a =
0.5 for lunch. To simplify the procedure for forecasting
for the SFSP in west Texas, an a of 0.5 was recommended.
The benefit of using a of 0.5 is that the new forecast could
be made by average the previous day's forecast and demand.
Appendix C illustrates the procedure to estimate the
forecast by using SES. The design of this worksheet
incorporated simplicity so it can be utilized without
computer aid. The cost and training for implementing the
SES method for this study will be minimal, but the benefits
of using this model may be significant.
Days of the week do not significantly influence the
forecast accuracy. Although there were some menu indices
indicating over-forecasting or under-forecasting, these two
factors, days of week and menu item, were not a common
problem overall. The likelihood of this event occurring in
this research setting was very minor. In consideration of
simplicity in applying a forecast method, it was recommended
that no adjustment be made to the menu index.
Impact of the Studv
The implication of this project is that food service
operators should consider and adapt forecasting methods as a
management tool. Whether under a non-profit or for-profit
84
Status, running a food service operation is a difficult and
increasingly more complex task. Forecasting helps to
control the two largest costs management addresses, labor
and food. Both of these expenses are variable and are a
function of customer demand, at least in part. Since the
margin between success and failure as a manager is tied to
how close an operation comes to achieving goals (profit
and/or staying within a budget), effective cost control is a
major concern. Successful operations do not meet or exceed
the expectations made of them by chance. Success happens
usually as a consequence of adhering to a specific plan.
Utilizing an effective forecast method will help any
manager function more effectively. This study highlights
the benefits and consequences of using forecasting. This
may be especially true in situations where managers want to
be able to use a forecast system but do not have the
training or expertise to deal with more sophisticated
forecasting techniques. Therefore, a simple forecast model
that is relatively accurate and simple to use would be more
desirable than a more complex model. Due to the short time
span of the SFSP operation, minimum training and low
expenses are imperative. The training cost for implementing
SES method should be minor. Therefore, SES is the best
method for CEI's sites to use for forecasting.
This study combined objective and subjective factors in
selecting a forecasting method. The researcher quantified
85
the subjective factors and combined them with the objective
factors and recommended the most appropriate method based on
quantitative analysis.
Limitation of the Results
This study recommended the simplest and most accurate
forecast method based on the aggregate data from different
sites of SFSP. The recommended method may not satisfy some
individual sites but is the most appropriate and desirable
method overall based on what was observed.
It is important to note the influence of time on
getting an accurate picture of the statistical strength and
predictive power of the SES forecasting method. The larger
the value of a, the quicker the forecast method will detect
the trend. A site has to have a trend present in their past
time plot in order to use a forecast method effectively. If
no trend is present, then the effectiveness of forecasting
by using SES is reduced. However, regardless of the pattern
that is present, SES should detect the pattern in a few
days. A food service operation might find that during the
first few days the forecast will deviate noticeably from the
actual demand; however, the greater the number of days a
food service operation is open, the more this forecast error
will flatten and, as a result, the forecast method's
accuracy will increase.
86
Recommendations for Further Research
This study analyzed only a small percentage of the SFSP
population in west Texas. Further research is recommended
in extending the scope of the research to other agencies who
operate SFSP in west Texas. Research also should be
conducted on larger SFSP operations. Since there are nearly
250 contractors for SFSP in Texas, many that operate much
larger programs than those in west Texas, future research
should include several different settings, such as a large
urban area, a medium sized city, and a small rural town. It
also would be to contrast the effectiveness of managing
operational resources by SFSP operations that used a
forecast method with SFSP that do not.
Moreover, a longitudinal study should explore the
impact of a forecast method on SFSP operations over several
years. The study could compare the forecasting
effectiveness for SFSP operations that used a forecast
method for one year, those that have used a forecast method
for two years, and those that have used it for five or more
years.
Summary
Because the challenges a manager faces have a major
impact on whether financial goals are reached, successful
management of a food service operation requires more
attention to the bottom line than ever before. What is
87
needed is a tool that helps reduce the guesswork and risk in
knowing the amount and types of food items that need to be
prepared for any given meal. Food wastage and superfluous
employee labor costs are very expensive mistakes that can
devastate any food service operation, for-profit or non
profit. Forecasting is a reliable and very effective way of
controlling these cost factors. It can readily be applied
to either a commercial or institutional setting, for profit
or non-profit, a large operation or a small one. Research
had demonstrated that forecasting, when used on a consistent
basis, can be an enormous help in aiding the manager to have
either a profitable operation or to stay within budget.
In summary, this research analyzed four forecast
methods, moving average (MA), simple exponential smoothing
(SES), Holt's exponential smoothing (Holt), and Winter's
exponential smoothing (Winter), and compared each method's
accuracy based on three criteria: pattern of time plot,
accuracy, and simplicity. The result of the comparison
shows that simple exponential smoothing outperforms the
others. It was determined that SES is not only the most
appropriate forecast method based on the subjective and
objective selecting criteria of forecast models, but also
the most accurate method. This research identified the most
accurate and most objective forecast method with minimal
cost and training requirements. The SES method may be
88
appropriate not only for the SFSP in west Texas but also
other short time span food service operations.
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APPENDIX A
SUMMER FOOD SERVICE PROGRAM FOOD PRODUCTION RECORD
Name of Site Date:
H in < PLJ
u: < CJ QC OQ
x: u 2 13 h ^
MENU
Q PJ m D
w Q O o
Q U W D
>H
H 2 rt:
e? 2 M > Ct
Ci] CO M CO
NUMBER SERVED
2
Q yA 1—1
in H h:i D Q <
S
(X o o CU
en EH H l :D
s Q ^ " (J - -o o ct: 2 CU M 1 >H
2 < O cu 2 - - TOTAL
94
APPENDIX B
DATA ANALYSIS SPREADSHEET
Codes of Data
The data entry includes ten columns. From the left to the right are column A through column J.
Column 1 SITE; a=Anson, c=Central, ck=Crockett, cw=Crowell, dz=DeZavala, fg=Floyd Gwin, g=Girls Inc., h=Hamlin, kc=Knox City, l=Lamar, m=Milam, s=South, t=Tulia, v=Vernon, and w=Woodson.
Column 2:
Column 3:
Column 4:
DATE; Date of service.
DAY; Day of week; l=Monday, 2=Tuesday, 3=Wednesday, 4=Thursday, and 5=Friday.
WEEK; Week of service; l=first week, 2=second week, etc.
Column 5: D ; Actual demand (children) of breakfast service.
Column 6 Dbn; Actual demand (non-program adult) of breakfast service.
Column 7: Di; Actual demand (children) of lunch service.
Column 8: Din; Actual demand (non-program adult) of lunch service.
Column 9
Column 10:
Mb; Breakfast entree; l=Bagel, 2=Cereal, 3=Donut, 4=Toast, 5=Cinnamon roll, 6=Grits, 7=0atmeal, 8=Muffin, 9=Pancake, 10=Eggs, ll=Bacon with bread, and 12=Breakfast Pizza.
Mi; Lunch entree; 1=BBQ, 2=Chicken nugget, 3=Corn Dog, 4=Hamburger, 5=Hot Dog, 6=Pizza, 7=Spaghetti, 8=Taco, 9=Ham & cheese sandwich, 10=Cheese sandwich, ll=Turkey sandwich, 12=Peanut butter & jelly sandwich, 13=Roast beef sandwich, 14=Tuna sandwich, 15=Burrito, 16=Steak finger, 17=Macaroni and cheese, 18=Bologna & soloni sandwich, 19=Fish nugget, 20=Chicken sandwich, and 21=Chicken salad sandwich.
95
APPENDIX C
ESTIMATED NUMBER OF MEALS FOR PREPARATION
WEEK OF: SITE
MEAL BREAKFAST or LUNCH
Calculate the number of CHILDREN'S meals to prepare for each day.
DAY NUMBER ESTIMATED
NUMBER SERVED
MONDAY ) ^ 2 = NEW ESTIMATE FOR TUESDAY
TUESDAY ) ^ 2 = NEW ESTIMATE FOR WEDNESDAY
WEDNESDAY ) ^ 2 = NEW ESTIMATE FOR THURSDAY
THURSDAY ) ^ 2 = NEW ESTIMATE FOR FRIDAY
FRIDAY ) ^ 2 =
NEW ESTIMATE FOR MONDAY
NOTE: THE VALUE OF THE ESTIMATE MAY NOT BE AS ACCURATE IN THE FIRST FEW DAYS OF SERVICE AS IT WILL BE AFTER THE FIRST WEEK.
96
APPENDIX D
SELECTED COLUMNS OF RAW DATA AND TIME PLOTS
Information in each site include:
SITE: The name of site being studied
YEAR: The year of observation
DATE column: Date of service
Db column: Breakfast demand
Di column: Lunch demand
Mb column: Breakfast menu; the menu codings were given on Appendix D.
Ml column: Lunch menu; the menu codings were given on Appendix D.
Chart on the top right: Time plot for breakfast with day of week coded
Chart on the bottom right: Time plot for lunch with day of week coded
Each point of a time plot indicates one observation of a meal. Coding of time plot is indicated on legend. Legend for time plot is as follows:
Monday = 1 or Mon. Tuesday = 2 or Tue. and etc.
Trend and seasonality values of each time plot were given on the bottom of each plot. The procedures of defining trend and seasonality values were given on Chapter III of this study.
97
98
S I T E : CENTRAL YEAR 1992
DATE Mb M,
4 - J u n 5 - J u n 8 - J \ i n 9 - J \ i n 1 0 - J u n 1 1 - J u n 1 2 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 1 8 - J i i n 1 9 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 5 - J u n 2 6 - J u n 2 9 - J i i n 3 0 - J u n 1 - J u l 2 - J u l 3 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 0 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 7 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 4 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 3 0 - J u l 3 1 - J u l 3 - A u g 4 - A u g 5 - A u g 6 - A u g 7 - A u g 1 0 - A u g 1 1 - A u g 1 2 - A u g 1 3 - A u g 1 4 - A u g
21 23 29 30 27 24 26 27 36 40 39 45 39 26 35 34 26 36 30 37 43 36* 35 24 34 30 24 23 23 34 23 22 35 19 26 36 23 23 24 34 43 24 22 22 40 22 28 33 32 44 30 26
53 57 70 72 93 86 86 84 80 106 96 106 80 70 84 82 88 88 94 101 94 93* 70 82 70 103 82 92 76 94 72 88 82 72 72 90 71 94 90 99 106 90 75 94 84 72 60 96 100 90 96 72
2 5 2 2 5 3 2 5 2 2 5 3 2 5 2 2 5 3 2 5 2
2 5 3 2 3 2 2 5 3 2 5 2 2 5 3 2 5 2 2 5 3 2 2 3 2 3 2 5 3 5
9 10 1 1 12 9 10 1 1 13 10 12 13 14 9 1 1 10 12 13 1 1 9 1 1 10
12 13 14 9 1 1 10 12 13 14 9 1 1 10 12 13 14 9 1 1 10 12 13 14 9 1 1 10 12 13 1 1 10 9 14
Connected Tine Plot Central Breakfast 199Z
49
+4
39
34
29
24
19
- • - • I • • • I • • • • I • • • ' I
l i i i i l a i i i J A i i i . J i — I I. i . ± , t ^ . J h ^ ^ i-.i u 1..1.
• Frl + non » THur • Tue
le 26 30 40 Se 66 Tine Series
Trend = 0 Seasonality = 1
Connected Tine Plot Central Lunch 1992
frl Hon THur
• Tue X yed
16 26 38 46 Tine Series
Trend = 0 Seasonality = 0
Estimated by SES method
99
SITE DeZAVALA YEAR: 1992
DATE M. M,
8-J\in 9-Jun 10-Jun 11-Jun 12-Jun 15-Jun 16-Jun 17-Jun 18-Jun 19-Jun 22-Jun 23-Jun 24-Jun 25-Jun 26-Jun 2 9-Jun 30-Jun 1-Jul 2-Jul 3-Jul 6-Jul 7-Jul 8-Jul 9-Jul 10-Jul 13-Jul 14-Jul 15-Jul 16-Jul 17-Jul 20-Jul 21-Jul 22-Jul 23-Jul 24-Jul
111 135 97 128 120 128 129 129 130 130 98 138 141 133 127 128 126 120 131 127* 130 127 131 125 129 124 125 120 127 129 124 98 89 101 62
265 243 265 276 259 279 304 303 302 302 250 310 319 298 301 280 290 306 312 301* 250 280 291 265 250 280 282 278 278 281 290 275 250 280 122
2 3 6 5 2 4 7 2 3 2 5 2 3 6 5 2 4 7 2
3 2 5 2 3 6 5 2 4 7 2 3 2 5 2
3 7 12 15 9 2 6 8 1 16 3 4 7 12 15 9 2 6 8
1 16 4 3 7 12 15 9 2 6 8 1 3 9 4
*Est imated by SES method
Connected Tine Plot DeZauala Breakfast 1992
142 -
• Fri + Hon • Thur • Tue X lied
16 26 38 Tine Series
46
Trend = 0 Seasonality = 0
Connected TiNe Plot DeZauala Lunch 199Z
326 -
288
248
288
168
128 •
• Fri + Hon * Thur • Tue X yed
16 26 36 Tine Series
46
Trend = 0 Seasonality = 0
100
S I T E : GIRLS INC. YEAR 1992
DATE Mv, Ml
4 - J u n 5 - J u n 8 - J u n 9 - J u n 1 0 - J \ i n 1 1 - J u n 1 2 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 1 8 - J u n 1 9 - J u n 2 2 - J v i n 2 3 - J u n 2 4 - J u n 2 5 - J u n 2 6 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 2 - J u l 3 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 0 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 7 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 4 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 3 0 - J u l 3 1 - J u l 3 - A u g 4 - A u g 5 - A u g 6 - A u g 7 - A u g 1 0 - A u g 1 1 - A u g 1 2 - A u g 1 3 - A u g 1 4 - A u g
57 42 55 64 63 64 55 4 5 5 5 69 72 4 9 52 66 66 69 62 55 69 82 68 69* 82 75 70 94 1 0 1 75 77 74 86 75 72 89 78 86 80 72 8 1 88 74 74 78 92 1 1 1 84 83 55 78 69 72 69
110 1 3 5 152 1 8 1 2 1 6 258 230 238 238 300 2 8 1 2 5 1 190 285 285 287 2 9 2 314 3 1 8 3 3 7 336 317* 3 0 7 3 1 7 3 1 9 264 2 7 0 2 2 7 2 6 5 3 1 9 3 0 3 2 6 0 3 0 3 258 3 1 8 3 1 9 2 5 7 2 6 7 2 6 7 2 9 2 3 1 7 292 2 8 0 3 3 2 274 3 1 3 2 6 0 2 2 7 258 240 2 6 8 2 2 8
7 5 2 3 2 5 2 2 4 2 3 7 2 5 6 2 3 7 2 5 2
3 2 2 2 3 6 2 3 4 7 2 3 2 5 7 3 6 4 7 5 7 2 3 6 2 6 3 4 6 7
15 4 3 7 12 15 9 2 6 8 1 16 4 3 7 12 15 6 2 8 9
1 16 4 3 7 12 15 9 2 6 8 1 16 4 3 7 12 15 9 2 17 8 1 16 4 7 12 3 6 9
Connected Tine Plot Girls Inc. Breakfast 1992
122
162
• Fri + Hon » Thur • Tue X Ued
Trend = 1 Seasonality = 0
Connected Tine Plot Girls Inc. Lunch 1992
• Fri + Hon » Thur a Tue X yed
16 26 36 48 58 66 Tine Series 1992
Trend = 0 Seasonality = 1
•Estimated by SES method
1 0 1
S I T E : SOUTH YEAR: 1992
DATE Db D, M>, Ml
15-Jun 16-Jun 17-Jun 18-Jun 19-Jun 22-Jun 23-Jun 24-Jun 25-J\m 26-Jun 29-Jun 30-Jun 1-Jul 2-Jul 3-Jul 6-Jul 7-Jul 8-Jul 9-Jul 10-Jul 13-Jul 14-Jul 15-Jul 16-Jul 17-Jul 20-Jul 21-Jul 22-Jul
77 108 95 144 100 107 112 125 117 183 139 169 135 143* 143* 133 135 123 125 127 121 139 129 131 123 103 94 107
238 331 338 339 326 313 363 358 410 338 344 391 406 376* 376* 386 371 383 402 361 330 385 365 385 397 354 321 326
7 3 2 3 2 5 6 2 3 5 2 4 7
3 2 5 2 3 6 5 2 4 7 7 2 4
2 6 8 1 16 4 15 3 7 12 9 2 6
1 16 4 3 7 12 15 9 2 6 16 1 8
*Est imated by SES method
Connected Tine Plot South Breakfast 1992
Fri
198
176
156
i 138 &
118
98
78
-.'....'.. ..;..'..!. ! •• ! ' '.1 . ' ' • • I • • • • ! ' • • • I
' • • • • * - • • • * ' . • . « - - . . I . . . . I
« Thur • Tue X yed
16 15 28 25 36
Tine Series
Trend = 1 Seasonality = 0
Connected Tine Plot South Lunch 1992
418
386
358
S 328
I 298
266
Z36
' * * * • ' * ' • • ' • • * * ' • • • • ' • ' ' • ' ' • ! * '
• Fri -•- Hon • Thur • Tue X yed
18 15 28
Tine Series 25 36
Trend = 1 Seasonality = 0
102
S I T E : TULIA YEAR 1 9 9 2
DATE Di My Ml
8 - J u n 9-Jxin 1 0 - J u n l l - J \ m 1 2 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 1 8 - J u n 1 9 - J u n 2 2 - J \ i n 2 3 - J u n 2 4 - J u n 25-Jvin 2 6 - J \ i n 2 9 - J u n 3 0 - J u n 1 - J u l 2 - J u l 3 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 0 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 7 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 4 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 3 0 - J u l 3 1 - J u l 3-Aug 4-Aug 5-Aug 6-Aug 7-Aug
15 20 27 23 18 41 43 50 49 56 54 52 65 61 54 45 54 48 52 52* 40 37 48 43 30 31 44 37 38 31 25 29 39 31 24 9 8 8 9 25 20 18 9 28 20
53 60 75 59 43 90 101 100 99 89 63 70 92 83 67 68 80 75 70 74* 44 69 58 72 55 56 61 58 60 46 47 51 40 50 33 30 45 43 35 12 33 44 42 50 55
2 5 2 5 3 2 5 2 2 3 2 2 3 2 5 2 5 2 2
2 5 2 2 3 2 5 2 2 3 2 5 3 2 5 2 5 2 2 3 2 5 2 3 3
18 12 10 9 18 18 12 14 10 18 18 12 10 14 18 9 11 10 12
9 11 10 1 14 9 11 10 12 18 9 11 10 12 14 9 11 10 12 14 9 11 14 12 18
Connected Tine Plot Tulia Breakfast 1992
• Fri + Hon « Thur • Tue X yed
Trend = 1 Seasonality = 0
Connected Tine Plot Tulia Lunch 1992
• Fri + Hon «< Thur • TUe X yed
28 38 Tine Series
•Estimated by SES method
Trend = 1 Seasonal i ty = 1
1 0 3
S I T E VERNON YEAR 1992
DATE D. Mv, M,
9 - J u n 1 0 - J u n 1 1 - J u n 1 2 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 1 8 - J \ i n 1 9 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 5 - J u n 2 6 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 2 - J u l 3 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 0 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 7 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 4 - J u l
59 58 5 5 6 5 4 3 50 6 5 4 9 50 4 5 64 4 7 60 5 5 4 7 5 8 6 1 4 2 52* 5 5 5 1 64 54 64 50 57 56 62 32 32 4 5 37 30 3 1
1 4 5 1 4 7 1 4 7 1 4 6 1 3 4 1 5 0 1 5 0 1 5 6 1 3 6 1 2 5 144 1 2 3 1 4 2 1 4 0 1 1 5 1 5 0 1 3 4 1 2 5 133* 1 4 4 1 4 3 144 1 3 3 1 2 6 1 3 1 152 1 4 4 1 4 4 1 3 1 1 1 0 1 1 0 1 0 6 1 1 4 1 0 0
2 2 5 2 5 2 3 6 5 2 3 6 2 3 5 2 3 7
5 2 3 7 5 2 4 7 2 3 5 3 2 2 3
9 10 2 7 15 3 4 2 6 9 6 12 4 1 15 9 16 6
4 3 7 15 12 9 2 6 8 1 4 7 12 15 9
*Est imated by SES method
Connected Tine Plot Uemon Breakfast 1992
• Frl + Hon * Thur a Tue X yed
Trend = 0 Seasonality = 1
Connected Tine Plot Uernon Lunch 1992
168
158
148
136
128
118
188
i " ^ ' ' ' ' r - ' ' - -T • • ^^—
J "+|f|j
'1 I > 1 1 1 J 1 1 1 L L 1 i - . J 1 1 1 1 1
r prrrr-, ,,,,,....,....,....,....
• Fri + tton « Thur a Tue X yed
16 26 36 Tine Series
46
Trend = 0 Seasonality = 0
104
S I T E : WOODSON YEAR 1 9 9 2
DATE Db M Ml
4 - J \ i n 5 - J u n 8 - J u n 9 - J u n 1 0 - J u n 1 1 - J u n 1 2 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 1 8 - J u n 1 9 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 5 - J u n 2 6 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 2 - J u l 3 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 0 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 7 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 4 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 3 0 - J u l 3 1 - J u l 3 - A u g 4 - A u g 5 - A u g 6 - A u g 7 - A u g
* E s t i m a t e d b y SES m e t h o d
8 13 20 21 16 32 22 12 14 30 25 26 28 27 29 30 22 40 25 31 32 30* 19 18 45 21 22 25 22 20 28 14 18 18 25 26 36 11 21 12 18 22 24 23 29 14 16
28 43 56 68 66 62 67 65 61 70 65 56 65 69 77 85 70 89 80 95 107 86* 80 80 97 113 94 70 80 78 80 76 78 89 103 125 90 78 78 88 86 97 95 101 94 80 80
2 5 2 2 5 3 2 5 2 2 5 3 2 5 2 2 5 3 2 5 2
2 5 3 2 5 2 2 5 3 2 5 2 2 5 3 2 5 2 2 5 3 2 5 2 2
9 9 10 12 13 11 9 11 10 12 13 14 9 1 1 10 12 13 14 9 1 1 10
12 13 14 9 1 1 10 12 13 14 9 1 1 10 12 13 14 9 11 10 12 13 9 14 1 1 10 12
Connected Tine Plot Uoodson Breakfast 1992
• Fri + Hon * Thur a Tue X yed
18 26 38 48 Tine Series 1992
58
T r e n d = 1 S e a s o n a l i t y = 1
Connected Tine Plot yoodson Lunch 1992
1 i
158
126
98
66
_ i 1 I I 1 ' ' • • '—1 I i . * > i I • • • • I I * . I * - i A i J . , i i
• Fri + Hon « Thur a Tue X yed
18 28 38 Tine Series
48 58
T r e n d = 1 S e a s o n a l i t y = 1
105
SITE ANSON YEAR: 1993
DATE Dv M>, M,
9-Jun 10-Jun 11-Jun 14-Jun 15-Jun 16-Jun 17-Jun 18-Jun 21-Jun 22-Jun 23-J\in 24-J\m 25-Jun 28-Jun 29-Jun 30-Jun 1-Jul 2-Jul 5-Jul 6-Jul 7-Jul 8-Jul 9-Jul 12-Jul 13-Jul 14-Jul 15-Jul 16-Jul 19-Jul 20-Jul 21-Jul 22-Jul 23-Jul
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
80 91 83 110 133 135 124 125 141 131 146 136 139 129 141 142 148 107 117* 101 127 135 122 91 104 115 112 81 95 100 98 94 90
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 7 15 2 6 8 15 19 20 16 15 3 7 15 2 6 8 1
20 16 4 3 7 15 2 6 8 1 19 20 15 3
•Es t imated by SES method
Connected Tine Plot Anson Lunch 1993
166
149 -
129 -
• 1 + Z m 3 a 4 X 5
199
19 29 39 Tine Series
Trend = 1 Seasonality = 1
106
SITE CENTRAL YEAR 1993
DATE M>, Ml
7 - J u n 8 - J u n 9 - J u n 1 0 - J u n 1 1 - J u n 1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 1 8 - J x i n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 5 - J u n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 2 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 6 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 3 0 - J u l 2 - A u g 3 - A u g 4 - A u g 5 - A u g 6 - A u g 9 - A u g 1 0 - A u g 1 1 - A u g 1 2 - A u g 1 3 - A u g 1 8
22 28 34 31 33 28 19 36 39 32 35 27 33 34 39 23 38 38 25 25 30* 24 24 28 27 23 35 35 32 31 22 38 36 35 40 34 30 40 23 32 21 52 34 20 23 27 37 31 29 28
49 93 97 112 117 118 94 118 103 95 98 83 103 106 112 94 106 120 120 120 114* 90 94 102 123 89 103 101 83 107 90 91 82 102 88 78 90 82 90 82 52 74 84 67 78 72 80 70 70 74
2 9 3 2 4 5 2 8 3 2 4 5 2 8 3 2 4 5 2 8
3 2 4 5 2 8 3 2 4 5 2 8 3 2 2 5 2 8 3 2 5 8 2 5 2 8 2 2 2
11 10 9 12 21 18 11 10 9 12 14 18 11 10 9 12 14 9 11 10
9 12 21 18 11 10 9 12 14 18 11 10 9 12 21 18 11 21 9 18 14 11 12 10 18 9 12 18
Connected Tine Plot Central Breakfast 1993
• Fri + Hon • Thur o Tue X yed
26 36 Tine Series
Trend Seasonality = 1
Connected Tine Plot Central Lunch 1993
129
169
• Fri + tton • Thur • Tue X yed
28 38 46 Tine Series
T r e n d = 1 S e a s o n a l i t y = 1
' E s t i m a t e d by SES method
1 0 7
S I T E CROCKETT YEAR: 1 9 9 3
DATE D^ M Ml
1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 1 8 - J u n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 5 - J u n 5 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 6 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 3 0 - J u l 2 - A u g 3 - A u g 4 - A u g 5 - A u g 6 - A u g 9 - A u g 1 0 - A u g 1 1 - A u g 1 2 - A u g 1 3 - A u g
102 156 1 8 9 1 3 1 176 2 0 3 2 0 5 2 0 1 2 0 5 2 0 4 196* 1 2 6 1 6 3 1 4 6 1 5 5 124 1 5 0 1 6 3 1 6 3 1 6 1 1 5 0 1 6 1 1 7 6 1 5 1 4 3 6 1 73 73 76 60 68 72 65 77 69 36 4 3 39 4 1 58
1 4 8 2 2 6 2 7 1 2 3 0 2 3 7 2 8 5 3 0 0 3 2 0 2 7 8 2 9 3 2 8 3 * 2 4 0 2 7 4 2 8 4 2 9 1 2 5 7 2 9 8 2 8 0 2 8 4 2 9 7 2 8 9 2 7 3 3 1 0 2 9 7 2 0 3 2 0 1 2 0 1 2 0 1 1 8 9 1 7 3 194 1 8 8 154 1 6 7 57 1 7 6 1 5 5 142 1 2 0 2 0 7
5 2 12 3 2 9 2 1 3 10
9 2 1 3 10 8 5 2 12 3 7 9 2 3 5 3 3 3 5 5 2 5 2 3 3 5 3 1 1
2 6 8 1 19 20 16 4 3 7
20 16 4 3 7 15 2 6 8 1 19 20 16 4 7 3 15 2 7 15 16 4 10 3 3 10 16 15 20
Connected Tine Plot Crockett Breakfast 1993
246 f-v
288
166
1 S 128
88
46
8
! ' • • • 1 ' • • • ! ' • • ' ) • - - I
? ; : •
I •
>
;
1/ \ ^ •
1 t i 1 i.. 1 1 I—1 1 1
1 2 3 4 S
Trend = 1
18 28 38 Tine Series
Seasonality = 0
48
Connected Tine Plot Crockett Lunch 1993
486 -
386
•Estimated by SES method
• 1 + 2 * 3 n 4 X 5
Trend = 1 Seasonality = 1
1 0 8
S I T E : CROWELL YEAR 1 9 9 3
DATE M. Ml
2 2 - J u n 2 3 - J u n 2 4 - J \ m 2 5 - J \ i n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 2 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 6 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 3 0 - J u l
1 1 9 9 1 1 9 13 15 13 13 13* 14 10 17 13 12 16 2 5 30 29 25 15 19 18 19 19 19 15 18 18
18 2 7 27 26 30 27 34 29 32 30* 2 5 30 33 3 5 34 50 58 63 50 4 5 4 6 4 3 4 0 59 3 5 48 44 30 30
* E s t i m a t e d b y SES m e t h o d
1 1 1 3 10 8 5 1 1 12 3
7 8 10 1 3 1 1 8 2 12 1 3 7 9 2 1 3 10 8 3
16 4 3 7 15 2 6 8 1
19 20 7 4 3 16 15 6 8 8 1 19 2 1 16 14 3 7 15 4
Connected Tine Plot CroMell BreaUast 1993
33
29
25
1
i 21
17
13
9
• 1 + 2 * 3 a 4 X 5
18 15 28 Tine Series
25 36
T r e n d = 0 S e a s o n a l i t y = 0
• 1 + 2 • 3 a 4 X 5
18 15 28 25 Tine Series
38
T r e n d = 1 S e a s o n a l i t y = 1
1 0 9
S I T E DeZAVALA YEAR: 1 9 9 3
DATE Mv, Ml
1 4 - J u n 1 5 - J x i n 1 6 - J u n 1 7 - J u n 1 8 - J u n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 5 - J u n 2 8 - J \ i n 2 9 - J u n 3 0 - J u n 1 - J u l 2 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 6 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 3 0 - J u l 2 - A u g 3 - A u g 4 - A u g 5 - A u g 6 - A u g 9 - A u g 1 0 - A u g 1 1 - A u g 1 2 - A u g 1 3 - A u g
65 72 92 96 87 1 1 5 1 0 1 92 93 1 0 1 99 1 0 9 95 1 0 1 80 94* 94 90 1 0 0 8 1 26 4 8 50 60 68 70 40 4 5 50 73 4 0 4 2 4 8 4 6 38 40 46 52 44 38 30 30 35 29 30
1 3 5 2 0 6 2 1 6 2 2 7 2 1 3 2 5 1 2 4 9 2 8 3 2 4 8 2 4 9 2 5 1 2 6 3 2 7 9 2 5 2 2 1 3 2 4 5 * 2 0 0 2 1 5 2 3 5 2 1 6 1 3 1 1 8 7 1 4 6 1 8 0 1 7 8 1 7 5 2 0 0 1 7 0 1 7 7 1 8 1 1 6 5 1 8 3 1 8 5 1 8 5 1 3 8 1 7 9 1 7 8 1 8 1 1 7 8 1 6 8 1 2 0 1 2 6 1 2 3 80 85
5 2 12 3 7 9 2 1 3 10 8 5 2 12 3
9 2 1 3 10 8 5 2 12 3 2 5 9 1 1 9 2 8 5 2 3 10 2 5 12 3 2 2 5 5
2 6 8 1 19 20 16 4 3 7 15 2 6 8 1
20 16 4 18 7 15 2 6 8 1 19 20 18 3 4 16 3 17 19 4 3 6 15 1 16 3 15 10 2
Connected Tine Plot DeZavala Breakfast 1993
• 1
+ 2 • 3 a 4 X 5
Trend = 1 Seasonality = 0
Connected Tine Plot DeZauala Lunch 1993
• 1 + 2 » 3 D 4 X 5
*Estimated by SES method
Trend = 1 Seasonality = 0
110
S I T E FLOYD GWIN YEAR 1 9 9 3
DATE M Ml
1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J \ m 1 8 - J u n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J x i n 2 5 - J u n 2 8 - J \ i n 2 9 - J u n 3 0 - J u n 1 - J u l 2 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 6 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 3 0 - J u l
• E s t i m a t e d b y SES m e t h o d
40 61 58 59 59 50 47 70 67 62 52 62 60 63 36 53* 43 65 55 67 50 50 52 46 54 63 57 51 52 49 50 64 61 70 45
100 120 109 109 116 123 123 120 114 113 121 130 145 127 112 123* 95 127 102 124 93 105 111 119 124 111 127 125 91 91 107 106 115 130 92
5 2 8 3 2 4 5 2 8 3 2 5 5 2 8
8 5 3 2 2 8 3 2 5 5 2 8 3 2 2 5 2 3 2
18 1 1 10 9 12 14 18 1 1 10 9 12 20 18 1 1 10
2 1 18 9 12 1 1 10 9 12 2 1 18 2 1 10 9 12 14 18 1 1 12 9
Connected Tine Plot Floyd Gwin Breakfast 1993
76
66
46
• 1 + 2 « 3 n 4 X 5
18 26 36
Tine Series 48
Trend = 0 Seasonality = 0
Connected Tine Plot Floyd Guin Lunch 1993
151
141
131
I-111
161
91
• 1 + 2 « 3 • 4 X 5
16 26 38
Tine Series 48
Trend = 0 Seasonality = 0
I l l
S I T E HAMLIN YEAR 1 9 9 3
DATE Mv, Ml
4 - J u n 7 - J u n 8 - J u n 9 - J u n 1 0 - J u n 1 1 - J u n 1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 1 8 - J \ i n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 5 - J u n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 2 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 6 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 3 0 - J u l 2 - A u g 3 - A u g 4 - A u g 5 - A u g 6 - A u g
62 62 65 68 68 61 72 59 53 52 48 49 51 53 50 54 54 52 49 16 17 34* 33 36 29 28 36 29 35 39 18 25 27 24 24 20 27 20 19 15 18 21 20 20 18 16
100 121 135 136 132 135 123 131 128 139 129 130 126 140 135 142 135 124 152 98 86 114* 101 107 105 96 84 97 111 106 93 95 117 123 118 97 106 94 101 101 103 99 105 106 105 99
2 9 1 3 10 8 2 2 1 3 7 9 2 1 3 10 8 5 2 1 3
9 2 1 3 10 8 5 2 1 3 7 9 2 1 3 10 8 2 8 3 2 2 12 3
16 17 4 3 7 15 2 6 8 1 19 20 16 4 3 7 15 2 6 8 1
20 16 4 3 7 15 2 6 8 1 19 20 16 4 3 7 15 16 7 15 7 8 16 7
Connected Tine Plot Hani in Breakfast 1993
• 1 + 2 m 3 D 4 X 5
18 28 38
Tine Series 46 58
Trend = 1 Seasonality = 1
Connected Tine Plot Hani in Lunch 1993
• 1 + 2 m 3 D 4 X 5
•Estimated by SES method
Trend = 0 Seasonality = 0
1 1 2
S I T E : KNOX CITY YEAR 1 9 9 3
DATE D>, Di Mv, Ml
9 - J u n 1 0 - J u n 1 1 - J u n 1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 1 8 - J u n 2 1 - J \ i n 2 2 - J \ i n 2 3 - J u n 2 4 - J u n 2 5 - J u n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 2 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l
3 1 46 44 40 50 43 4 9 55 34 44 44 40 42 53 56 54 55 53 52* 55 53 52 53 54 50 53 50 57 50 54 35 50 30
59 73 74 76 9 1 88 98 99 85 9 1 94 9 1 90 107 1 0 5 1 1 5 116 1 1 7 110* 1 0 5 107 110 100 1 0 1 102 1 0 5 106 110 1 0 1 110 74 60 70
* E s t i m a t e d b y SES m e t h o d
3 10 8 3 2 12 3 7 9 2 1 3 10 8 5 2 12 3
9 2 1 3 10 8 5 2 12 3 7 9 2 1
3 7 15 2 6 8 1 19 20 16 4 3 7 15 2 6 8 1
20 16 4 3 7 15 2 6 8 1 19 20 16 10
Connected Tine Plot Knox City Breakfast 1993
• 1 + 2 m 3 a 4 X 5
18 26 36 Tine Series
T r e n d = 0 S e a s o n a l i t y = 0
Connected Tine Plot Knox City Unch 1993
• 1 + 2 m 3 a 4 X 5
18 26 38 Tine Series
T r e n d = 0 S e a s o n a l i t y = 0
1 1 3
SITE LAMAR YEAR 1 9 9 3
DATE M, Ml
1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 1 8 - J \ i n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 5 - J \ i n 2 8 - J u n 2 9-Jun 30-Jun 1-Jul 2-Jul 5-Jul 6-Jul 7-Jul 8-Jul 9-Jul 12-Jul 13-Jul 14-Jul 15-Jul 16-Jul 26-Jul 27-Jul 28-Jul 29-Jul 3 0-Jul 2-Aug 3-Aug 4-Aug 5-Aug 6-Aug 9-Aug 10-Aug 11-Aug 12-Aug 13-Aug
42 70 77 73 81 81 85 89 86 88 88 86 82 86 85* 85. 87 81 90 87 85 87 84 88 88 67 66 50 50 60 47 5 10 20 30 20 26 28 15 27
99 109 133 140 134 122 188 208 205 208 210 210 206 210 203* 203* 183 183 198 205 187 208 210 207 206 200 158 194 205 200 197 187 205 210 165 184 195 200 215 207
5 2 12 3 7 9 2 3 1 10 8 5 2 12
2 9 1 3 10 8 5 2 12 2 7 3 1 1 5 2 4 2 9 2 9 2 4 2
2 6 8 1 19 21 16 3 4 7 15 2 6 8
16 20 4 3 7 15 2 6 8 20 7 3 1 10 2 7 15 8 3 2 15 6 15 6
*Estimated by SES method
Connected Tine Plot Lanar Brealcfast 1993
« K V
166
88
68
48
28
6
• 1 + 2 « 3 a 4 X 5
16 28 38 Tine Series
48
Trend = 0 Seasonality = 1
Connected Tine Plot Lanar Lunch 1993
• 1 + 2 « 3 n 4 X 5
16 28 36
Tine Series
Trend = 0 Seasonality = 0
114
S I T E : TULIA YEAR: 1 9 9 3
DATE Dy Di M Ml
8 - J u n 9 - J u n 1 0 - J u n 1 1 - J u n 1 4 - J \ i n 1 5 - J u n 1 6 - J u n 1 7 - J u n 1 8 - J u n 2 1 - J u n 2 2 - J \ i n 2 3 - J u n 2 4 - J \ i n 2 5 - J i i n 5 - J u l 6 - J u l 7 - J u l 8 - J u l 9 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 6 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 3 - J u l 2 6 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 3 0 - J u l 2 - A u g 3 - A u g 4 - A u g 5 - A u g 6 - A u g
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
58 68 93 106 96 144 142 132 158 163 165 165 165 165 159* 143 150 150 147 138 131 144 118 14 6 111 133 134 148 136 107 119 131 147 127 93 102 115 107 70
0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
• E s t i m a t e d b y SES m e t h o d
16 4 3 7 2 6 8 1 19 2 16 4 3 7
20 16 4 3 7 15 2 6 8 1 19 20 16 4 15 7 1 6 15 8 1 19 3 12
Connected Tine Plot Tulia Lunch 1993
178
156
138
U 8
96
76
56
: / ^
• 1 + 2 • 3 o 4 X 5
16 28 36 Tine Se r i e s
48
T r e n d = 1 S e a s o n a l i t y = 1
115
S I T E : CENTRAL YEAR 1 9 9 4
DATE Mb Ml
6 - J u n 7 - J u n 8 - J u n 9 - J u n 1 0 - J u n 1 3 - J \ i n 1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 2 0 - J u n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 7 - J u n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 4 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 1 1 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 8 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 5 - J u l 2 6 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l 1 -Aug 2 - A u g 3 - A u g 4 - A u g 5 - A u g 8 - A u g 9 - A u g 1 0 - A u g 1 1 - A u g 1 2 - A u g 1 5 - A u g 1 6 - A u g 1 7 - A u g 1 8 - A u g 1 9 - A u g
5 13 8 17 9 10 15 10 12 18 8 6 15 16 14 7 14 11 9 14 12* 4 14 14 17 3 9 9 10 10 12 14 11 13 8 8 21 10 11 9 13 12 17 20 10 16 16 16 17 13 18 11 13 14 21
46 54 58 40 43 30 29 35 24 24 34 30 43 40 44 38 42 45 41 46 42* 39 34 54 33 24 41 36 42 44 38 31 41 32 42 38 42 34 28 29 34 37 42 48 34 35 35 44 44 28 41 35 41 42 44
2 1 8 3 2 8 1 2 1 8 3 2 8 1 2 1 8 3 2 8
1 2 1 8 3 2 8 1 2 1 8 3 2 8 1 2 1 8 3 2 3 1 2 1 3 2 1 3 2 2 3 2 3 2
10 12 2 1 18 14 9 10 18 12 2 1 18 14 9 10 14 12 2 1 18 14 9
10 1 1 12 2 1 18 14 9 10 10 12 2 1 18 14 9 10 10 12 2 1 18 14 9 10 18 12 2 1 18 14 9 10 18 12 2 1 10 14
Connected Tine Plot Central Breakfast 1994
e
• 1 + 2 « 3 o 4 X 5
T r e n d = 0 S e a s o n a l i t y = 1
Connected Tine Plot Central Lunch 1994
64
54 -
1
34
24
• 1 + 2 « 3 a 4 X 5
T r e n d = 0 S e a s o n a l i t y = 1
• E s t i m a t e d b y SES m e t h o d
116
S I T E DeZAVALA YEAR 1 9 9 4
DATE M, Ml
6 - J u n 7 - J u n 8 - J u n 9 - J u n 1 0 - J u n 1 3 - J u n 1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 2 0 - J u n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 7 - J u n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 4 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 1 1 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 8 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l
'Estimated by SES method
52 87 85 77 77 64 77 57 80 77 60 64 75 75 72 72 70 67 65 24 55* 21 19 16 11 12 16 10 12 16 3 9 4 8 12
192 237 281 262 275 288 263 275 299 276 291 311 335 334 301 263 299 337 330 301 311* 228 234 190 236 200 239 186 196 231 187 187 186 154 200
2 1 4 3 2 3 4 2 1 4 3 2 4 3 2 1 4 3 4 3
3 2 1 3 3 2 2 2 2 2 2 2 2 2
7 3 8 4 2 6 1 7 3 4 4 2 8 3 7 3 8 4 3 6
3 7 3 4 8 4 3 7 6 4 8 3 3 2
Connected Tine Plot DeZavala Breakfast 1994
• 1 + 2 « 3 n 4 X 5
Trend
19 26 39 Tine Series
Seasonality = 0
49
Connected Tine Plot DeZauala Lunch 1994
e e c u
359
319 -
279 -
239
199 -
159
1 2 3 4 5
19 29 39
Tine Series
Trend = 0 Seasonality = 0
1 1 7
S I T E FLOYD GWIN YEAR 1 9 9 4
DATE Mv, Ml
1 3 - J u n 1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 2 0 - J u n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 7 - J u n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 4 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 1 1 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 8 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 5 - J u l 2 6 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l
' E s t i m a t e d b y SES m e t h o d
55 43 48 27 52 26 45 64 40 57 30 30 50 34 44 41* 40 46 48 44 46 47 46 47 49 32 46 46 44 47 41 34 29 29 21
82 102 98 102 101 68 104 103 77 72 68 100 80 74 78 80* 94 108 109 91 108 125 110 112 106 100 114 112 68 112 75 82 87 70 77
8 1 2 1 8 3 2 8 1 2 1 8 3 2 8
1 2 1 8 3 2 8 1 2 1 8 3 2 8 1 2 1 8 3
10 10 18 12 2 1 18 14 9 10 18 12 2 1 18 14 9
10 1 1 12 2 1 18 14 9 10 1 1 12 2 1 18 14 9 10 10 12 2 1 18
Connected Tii«e Plot Floyd Gwin Breakfast 1994
i
• 1 + 2 » 3 • 4 X 5
19 29 39 Tine Series
T r e n d = 0 S e a s o n a l i t y = 1
Connected Tine Plot Floyd Gwin Lunch 1994
128 -
• 1 + 2 m 3 a 4 X 5
19 29 36 Tine Sereis
T r e n d = 0 S e a s o n a l i t y = 1
118
S I T E : HAMLIN YEAR 1994
DATE Dv Di Mb Ml
6 - J u n 7 - J u n 8 - J u n 9 - J u n 1 0 - J u n 1 3 - J u n 1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J v i n 2 0 - J u n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 7 - J u n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 4 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 1 1 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 8 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 5 - J u l 2 6 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l
40 53 44 53 46 48 45 40 40 30 47 41 49 47 48 38 47 49 35 25 37* 25 34 25 36 32 37 31 30 29 29 34 29 31 34 30 30 30 28 30
94 108 110 118 117 115 118 117 110 114 118 119 121 117 122 113 119 127 118 90 111* 93 95 93 97 84 91 90 91 81 86 95 99 91 81 81 83 88 93 93
2 1 4 3 2 3 4 2 1 4 3 4 1 2 3 3 2 4 2 1
4 3 2 3 1 4 2 4 3 2 3 4 2 7 2 4 3 7 4
• E s t i m a t e d by SES me thod
7 3 8 4 2 6 1 7 3 8 2 1 6 7 4 8 1 4 6 3
8 4 2 6 3 3 7 8 4 2 6 3 7 3 4 7 2 6 3
Connected Tine Plot Hani in Breakfast 1994
• 1 + 2 * 3 n 4 X 5
Trend = 0 Seasonality = 0
Connected Tine Plot Hani in Lunch 1994
I
131
121 -
111
• 1 + 2 » 3 n 4 X 5
191
19 29 39 Tine Series
Trend = 0 Seasonality = 0
119
S I T E KNOX CITY YEAR 1 9 9 4
DATE D i M, Ml
1 - J u n 2 - J u n 3 - J u n 6 - J u n 7 - J u n 8 - J u n 9 - Jv in 1 0 - J u n 1 3 - J u n 1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 2 0 - J \ i n 2 1 - J u n 2 2 - J u n 2 3 - J \ i n 2 4 - J u n 2 7 - J u n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 4 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 1 1 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 8 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l
• E s t i m a t e d b y SES m e t h o d
23 16 22 52 56 60 54 52 54 54 55 57 55 42 53 52 54 51 49 53 50 50 47 50* 30 27 28 28 25 26 26 27 24 26 25 23 20 22
50 48 55 99 91 101 98 100 92 96 100 98 97 93 100 99 100 102 101 102 98 99 100 100* 75 72 70 55 62 61 64 66 61 63 60 58 55 60
2 1 3 2 3 4 2 1 4 3 2 3 4 2 1 4 3 2 3 4 2 1 4
3 2 3 4 2 1 4 3 2 3 4 2 1 4
7 3 4 2 6 1 7 3 8 4 2 6 3 7 3 6 4 2 6 3 7 3 8
4 2 6 3 7 3 8 4 2 6 3 7 3 8
Connected Tine Plot Knox City Breakfast 1994
• 1 + 2 * 3 • 4 X 5
T r e n d = 1 S e a s o n a l i t y = 0
Connected Tine Plot Knox City Lunch 1994
• 1 + 2 • 3 a 4 X 5
19 29 39 Tine Series
T r e n d = 0 S e a s o n a l i t y = 0
1 2 0
S I T E LAMAR YEAR: 1994
DATE Di Mv, M,
6 - J u n 7 - J u n 8 - J u n 9 - J u n 1 0 - J u n 1 3 - J u n 1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 2 0 - J u n 2 1 - J u n 2 2 - J u n 2 3 - J \ i n 2 4 - J u n 2 7 - J u n 2 8 - J u n 2 9 - J u n 30-J \xn 1 - J u l 4 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 1 1 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 8 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l
40 62 63 58 53 36 40 45 60 51 50 55 53 50 47 56 47 60 50 5 38* 4 14 12 12 14 15 20 17 14 16 5 4 4 5
168 173 211 212 163 189 224 197 209 195 209 218 216 178 210 218 219 229 207 135 190* 144 160 140 110 150 84 82 80 79 74 95 89 42 45
2 1 2 3 3 4 4 2 1 3 2 3 2 2 2 3 4 2 1 3
3 2 2 1 2 4 4 3 2 2 3 3 2 3
•Estimated by SES method
7 3 2 4 6 8 1 7 3 4 2 6 8 7 3 4 3 2 8 3
6 4 7 3 2 8 3 6 3 4 2 3 6 2
Connected Tine Plot Lanar Brealcfast 1994
• 1 * 2 » 3 a 4 X 5
Trend
19 26 39 Tine Series
Seasonali ty = 1
Connected Tine Plot Lanar Lunch 1994
249 -
1 2 3 4 5
19 29 39 Tine Series
Trend = 1 Seasonal i ty = 1
1 2 1
S I T E MILAM YEAR: 1994
DATE Mv, Ml
6 - J u n 7 - J u n 8 - J u n 9 - J u n 1 0 - J u n 1 3 - J u n 1 4 - J u n 1 5 - J u n 1 6 - J \ i n 1 7 - J u n 2 0 - J u n 2 1 - J u n 2 2 - J i i n 2 3 - J \ i n 2 4 - J u n 2 7 - J u n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 4 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 1 1 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 8 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l
*Estimated by SES method
23 45 41 45 45 35 28 24 40 23 24 38 36 33 33 27 36 28 42 8 27* 8 6 14 12 7 14 12 13 11 7 9 7 6 3
103 151 187 201 167 146 182 167 190 97 150 201 153 207 171 170 149 198 185 115 159* 120 113 123 102 103 123 113 127 90 78 118 93 135 78
2 1 2 3 4 4 2 1 3 2 4 3 4 2 1 4 3 2 3 4
1 2 4 3 2 3 4 2 1 4 3 2 1 4
7 3 2 6 4 1 7 3 4 2 8 6 1 7 3 8 4 2 6 1
3 7 8 4 2 6 3 7 3 8 4 2 7 1
Connected Tine Plot tlilan Breakfast 1994
• 1 + 2 * 3 • 4 X 5
19 29 39 Tine Series
T r e n d = 0 S e a s o n a l i t y = 1
Connected Tine Plot Hi Ian Lunch 1994
229 -
199 -
i 169
• 1 + 2 » 3 a 4 X 5
139 -
196 -
19 29 39
Tine Series
Trend = 0 Seasonality = 1
122
S I T E STANTON YEAR: 1994
DATE Db M Ml
1 - J u n 2 - J u n 3 - J u n 6 - J u n 7 - J u n 8 - J u n 9 - J u n 1 0 - J u n 1 3 - J u n 1 4 - J u n 1 5 - J u n 1 6 - J u n 1 7 - J u n 2 0 - J u n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 7 - J u n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 4 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 1 1 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 8 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l 2 5 - J u l 2 6 - J u l 2 7 - J u l 2 8 - J u l 2 9 - J u l
* E s t i m a t e d by SES method
26 70 69 46 69 74 61 60 55 71 61 53 67 47 50 57 54 50 50 60 48 55 46 51* 41 38 40 42 40 42 43 54 50 30 50 41 48 33 40 32 32 30 31
81 112 98 105 146 151 144 149 130 150 148 154 145 150 151 144 138 137 151 157 146 171 144 152* 138 148 129 126 138 127 137 141 137 130 130 122 123 128 126 121 101 126 120
2 1 4 3 2 3 7 2 1 4 3 2 3 4 2 1 4 3 2 3 4 2 1
4 3 2 3 4 2 1 4 3 2 3 2 3 2 1 2 2 2 3
7 3 8 4 2 6 1 7 3 6 4 2 3 6 7 3 8 4 2 6 4 7 3
8 4 2 6 3 7 3 8 4 2 6 3 7 3 6 4 2 6 3
Connected Tine Plot Stanton Breakfast 1994
7 2
• 1 + 2 « 3 a 4 X 5
29 39
Tine Series
Trend = 1 Seasonality = 0
Connected Tine Plot Stanton Lunch 1994
181
161
141 -
1 2 3 4 5
121
191
19 29 39
Tine Series
49
Trend = 1 Seasonality = 0
123
S I T E : TULIA YEAR 1 9 9 4
DATE Dv, Di Mv, Ml
13-Jun 14-Jun 15-Jun 16 -Jun 17-Jun 20-J\in 21-Jun 22-Jun 23-Jun 24-Jun 27-Jun 28-Jun 29-Jun 30-Jun 1-Jul 4-Jul 5-Jul 6-Jul 7-Jul 8-Jul 11-Jul 12-Jul 13-Jul 14-Jul 15-Jul 18-Jul 19-Jul 20-Jul 21-Jul 22-Jul
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
167 173 180 205 171 174 154 157 170 162 164 181 179 176 132 161* 100 111 127 105 108 125 115 101 81 87 82 96 96 91
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 0 0
7 3 8 4 2 6 3 1 7 3 8 4 2 6 3
7 3 8 4 2 6 3 7 3 8 4 2 6 3
"Estimated by SES method
Connected Tine Plot Tulia Unch 1994
239
299
179
149
119
89
T ' • • ' ! ' • • ' 1 ' • ' ' ! ' • ' • 1 • • ' ' 1
1 1 I i 1 1 , , , . 1 1 > • 1 1 • • 1 1
• • • «
1 . . . .
-
-
1
• 1 + 2 • 3 a 4 X 5
19 15 29 25 39
Tine Series
Trend = 1 Seasonality = 1
124
S I T E : VERNON YEAR 1994
DATE Di Mv, Ml
6 - J u n 7 - J u n 8 - J u n 9 - J u n 1 0 - J u n 1 3 - J u n 1 4 - J \ i n 1 5 - J x i n 1 6 - J u n 1 7 - J \ i n 2 0 - J i i n 2 1 - J u n 2 2 - J u n 2 3 - J u n 2 4 - J u n 2 7 - J u n 2 8 - J u n 2 9 - J u n 3 0 - J u n 1 - J u l 4 - J u l 5 - J u l 6 - J u l 7 - J u l 8 - J u l 1 1 - J u l 1 2 - J u l 1 3 - J u l 1 4 - J u l 1 5 - J u l 1 8 - J u l 1 9 - J u l 2 0 - J u l 2 1 - J u l 2 2 - J u l
62 70 59 70 54 62 70 52 55 50 59 56 63 61 55 56 66 54 48 43 52* 40 65 50 47 37 39 37 41 35 38 35 29 29 33
125 133 107 113 114 143 141 128 123 96 125 127 132 115 95 122 124 160 123 111 123* 94 113 104 96 105 93 94 90 80 81 91 78 84 93
2 1 4 3 2 3 4 2 1 4 3 2 3 2 1 4 4 3 2 3
4 3 1 4 3 2 3 4 2 1 4 1 2 1
*Estimated by SES method
7 3 8 4 2 6 1 7 3 1 4 2 6 7 3 8 5 4 2 6
5 7 3 8 4 2 6 5 7 3 8 4 6 5
Connected Tine Plot Uernon Breakfast 1994
• 1 + 2 » 3 • 4 X 5
19 29 39
Tine Series
Trend = 1 Seasonality = 0
Connected Tine Plot Uernon Lunch 1994
i
178
158 -
138 -
• 1 + 2 » 3 a 4 X 5
118
19 29 39
Tine Series
Trend = 1 Seasonality
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