A DISCRETE STATE MODEL
FOR SCHEDUUNG ONCErOVER

. HARVEST or 910mm; CUCUMBERS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNEVERSITY
ASHOK KUMAR PATEL
1973

 

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ABSTRACT

A DISCRETE STATE MODEL FOR SCHEDULING ONCE-OVER
HARVEST OF PICKLING CUCUMBERS

BY

Ashok Kumar Patel

The general problem of scheduling formed the basis
of the thesis. In particular a cucumber fruit development
model was developed to assist the manager in making a
harvest time decision for once-over mechanical harvest.

The application of the model for the grower is in making
reliable forecasts of the harvest data of a field 2-3 days
in advance to Optimize his expected gains. The application
of the model for the processor includes scheduling to reduce
field losses due to unforseen shortages of equipment and/or
labor and regulation of the flow of desired size distribu-
tion mix into the plant.

The model prOposed is a class of general linear
dynamic models of discrete time and constant parameter form.
The model can be viewed in light of discrete state space
theory or population dynamics. Both views are helpful.

The states of the model are the number of fruit of various
size grades at time T. The output from the model is the
weight of fruit of various size grades at time T. The

input to the model is a unit step that corresponds to a
constant number of new fruit of smallest size grade entering
into the system each day. The parameters in the model were.

estimated using the technique of multiple regression with

Ashok Kumar Patel

least squares criterion.

Model verification was attempted using both the
available statistical tests as well as analyses of the
time behavior from the known historical record. Model
verification along with sensitivity analysis has proved the
general adequacy of the model in making harvest time deci-
sions for different locations and seasons.

The model was implemented for on-line applications
via a teletype-telephone-time share computer hook-up.

The following general conclusions can be made:

(1) The model proposed for cucumber fruit development
is valid under the assumptions made in its develop—
ment.

(2) The model shows promise of being a valuable tool
in accurately scheduling 2-3 days in advance the
optimal harvest date.

(3) The model implementation is feasible economically

and operationally.

Approved ”/3 ALQC...

Mfijor’Professor

9 K MA...“

Depaftment Chairman

A DISCRETE STATE MODEL FOR SCHEDULING ONCE-OVER

HARVEST OF PICKLING CUCUMBERS

BY

Ashok Kumar Patel

A THESIS

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Engineering

1973

‘u ACKNOWLEDGMENTS

The author wishes to express his appreciation to:
Dr. J. B. Holtman for his guidance, encouragement and sup-
port throughout the course of the thesis.

Dr. J. B. Kreer, Dr. T. J. Manetsch and Dr. Habib
Sakhi for their suggestions, constructive criticisms, and
review of the thesis.

The National Pickle Grower's Association and Pickle
Packer's International for providing financial support.

Daley Pickles, Saginaw, Michigan for providing the
data utilized in the study.

The staff of the Department of Agricultural Economics

for their help.

ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . .

LIST OF TABLES O O O O O O O O O 0

LIST OF FIGURES . . . . . . . . .

Chapter

1.

INTRODUCTION . . . . . . .

1.1 Purpose of the Study .
1.2 Sc0pe of the Study . .
1.3 Objective . . . . . .

REVIEW OF LITERATURE . . .

2.1 Factors Affecting Rate
2.2 Factors Affecting Rate
of Cucumbers . . . .
2.2.1 Temperature .

2.2.2 Soil Moisture

2.2.3 Plant Population

2.2.4 Fertilizers . . .
2.3 Rate of

MODEL DEVELOPMENT . . . . .

3.1 Existing Approaches . . . . .
3.2 An Alternative Approach . . .
3.3 Data Collection . . . . . . .
3.4 Model Formulation . . . . . .
3.4.1 Available Inputs . .
3.4.2 Desired Outputs . . .
3.4.3 Model Structure . . .
3.4.4 Time Period . . . . .
3.5 Functional Relationships . . .
3.6 Parameter Estimation Procedure

3.6.1 Stepwise Addition of Variables
3. 6. 2 Stepwise Deletion of Variables

iii

Growth Curves for Cucur b

of Fruit Growth
of Growth

it Fru

Page

ii

vi

quot—0 H

10
11
11
12
13

15

15
17
22
24
24
24
24
26
27
33
34
35

Chapter Page
3.7 Estimated Relationships . . . . . . . . . 36
3 O 8 Estimated MOdels O 0 O O 0 O O O O I O O O 44
3 O 9 A Simulation MOdel O O O O O O O O O O O O 46

4. MODEL VERIFICATION . . . . . . . . . . . . . . 51

4.1 Statistical Tests . . . . . . . . . . . . 51
4.1.1 Model Structure . . . . . . . . . 51
4.1.2 Test for Autocorrelation . . . . 53
4.2 Time Behavior . . . . . . . . . . . . . . 56

5. SENSITIVITY ANALYSIS . . . . . . . . . . . . . 69
6. LINEAR DYNAMIC MODELS . . . . . . . . . . . . . 80

6.1 Analytic Solution . . . . . . . . . . . . 81
7. CASH VALUE PREDICTION . . . . . . . . . . . . . 84

7.1 Aggregate Price Structure . . . . . . . . 84
7.2 Differential Price Structure . . . . . . . 85

8. USE OF THE MODEL . . . . . . . . . . . . . . . 95

9. IMPLEMENTATION . . . . . . . . . . . . . . . . 97
10. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . 99
11. RECOMMENDATIONS . . . . . . . . . . . . . . . . 101
REFERENCES . . . . . . . . . . . . . . . . . . . . . . 102
APPENDICES . . . . . . . . . . . . . . . . . . . . . . 106
APPENDIX A . . . . . . . . . . . . . . . . . . 106

APPENDIX B O O O O O O O O O O O O O O O O O O 108
APPENDIX C O O O O O O O O O O O O O O O O O O 109

iv

10.

LIST OF TABLES

Page
Two typical fruit deve10pment patterns . . . . 4
Durbin-Watson test statistics (d) . . . . . . 55
Mean and standard deviations of percentage
error over the range of all sequences . . . . 66
Mean and standard deviation of fruit of size
grade four for different locations and
seasons . . . . . . . . . . . . . . . . . . . 73
The results of parametric simulation . . . . . 77
Means and standard deviations of percentage
error for one day forecast for 1971 season . . 90
Means and standard deviations of percentage
error for one day forecast for 1972 season . . 91
Means and standard deviations of percentage
error for two day forecasts . . . . . . . . . 92
Means and standard deviations of percentage
error for forecast up to 5 days for 1971
season 0 O O O O O I O O O O O I O O O O O O O 93
Means and standard deviations of percentage
error for forecast up to 5 days for 1972
season 0 O O O O O O O O O O O O O O O O O O O 94

LIST OF FIGURES

Figure Page
1. A schematic diagram of the cucumber
number-class structure . . . . . . . . . . . . 25
2. Actual and simulated values for size grade 4 . 57
’ 3. Actual and simulated values for size grade 3B 58
4. Actual and simulated values for size grade 3A 59
5. Actual and simulated values for size grade 2 60
6. Actual and simulated values for size grade 1B 61
7. Actual and simulated values for size grade 1A 62
8. Actual and simulated values for total number

and total weight . . . . . . . . . . . . . . . 63

9. Linear constant-parameter dynamic model . . . 82
10. Expected cash value behavior . . . . . . . . . 86
ll. Useable weight behavior . . . . . . . . . . . 88

vi

1 . INTRODUCTION

Michigan is the leading state in the production
of cucumbers for pickling, producing 25 to 85 per cent of
the nation's total. The cucumber is the most important
processing vegetable grown in Michigan, representing 60
per cent of the acreage and 50 per cent of the value of
processing vegetable crops (1). Pickling cucumbers also
account for a significant part of the vegetable processing

industry in Wisconsin, California and North Carolina.

1.1 Purpose of the Study

 

The trend toward growing more cucumbers strictly
for processing, the rising cost of farm labor and its
relatively uncertain availability in the past few years,
coupled with a need to increase grower income and worker
productivity all contribute to the need for the mechanical
harvest of this crop. As a result, in just a few years
mechanical harvest of pickling cucumbers has increased
from zero to a substantial majority of all acreage. In
1968 mechanical harvesters harvested approximately 20 per
cent of the Michigan acreage. Estimates place the 1972
acreage mechanically harvested at 90% (2).

Initial attempts to develop a multi-pick cucumber

harvester failed primarily because of their poor and
inconsistent performance (3). This led to the development
of machines for destructive or once-over harvest. In 1964
Stout et a1. (3) described a harvester prototype using

the constriction principle for fruit removal combined with
a mechanism to convey plants to the rollers. The fruit
were conveyed to containers and the vines discharged.
Commercial production of this machine, with modifications,
began in 1964.

The use of once-over machines solved some of the
earlier problems of high labor costs and decreasing
productivity. However, it gave birth to entirely new
problems of decreasing quantity as well as quality of
product in relation to the hand harvested product.

Research conducted by Cargill and associates in 1969 (2)
indicated that hand harvested production averages approxi—
mately 200 bushels per acre; whereas once over machine
harvested production averages approximately 100 bushels

per acre--a 50% reduction. The same research also revealed
that both the quantity as well as the quality of the product
delivered by present mechanical systems of harvesting,
handling and unloading is dependent upon many factors among
which physical characteristics of the equipment and the

time of harvest are very critical. The latter is the

subject matter of this thesis.

1.2 Scope of the Study

 

Time of harvest influences not only the grower's
income and the processor's raw product size distribution,
but it has also been found to be related to such complicating
factors as the presence of the number of stems and the
extent of damage in mechanically harvested cucumbers. The
study conducted by Cargill and associates (2) showed that
the cucumbers harvested in the afternoon had more stems and
less mechanical injury than cucumbers harvested in the
morning.

Table 1 shows two examples of the variation in cash
value of fields of cucumbers over time, utilizing a common
price schedule. Other price schedules produce similar levels
of variability. These trajectories are not atypical. In
24 hours the value of a field typically changes 5 to 10 per
cent and sometimes as much as 20 per cent or more. Daily
increase in weight ranges from 10 to 30 per cent but under
favorable conditions of growth, it may be as high as 40 per
cent or more. It has been observed (4) that the maturity
change is rapid with pickling cucumbers and a field may go
from optimum maturity to overmaturity or culls, in just a
day or two. In hot weather with adequate soil moisture, the
period for optimum harvest may be as brief as 12 hours. As
suggested by Table l the maximum yield and the maximum value
of the crop does not occur at the same time. Packers often

differ in their requirements for Cucumber size grades.

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Therefore, the grower and the processor usually agree before
the growing season on the system of payment.

The harvest scheduling problem is further compli-
cated by the need for a forecast of the optimum harvest
date one to two days in advance of the Optimum date. This
information is required for accurate planning Of field,
transport and plant processing operations. Sims and Zahara
(5) suggest that the entire harvesting operation should be
considered as one coordinated operation, involving the fruit
to be harvested, personnel for sorting, trucking equipment
to receive and to transport the fruit, fork lifts, containers
for the fruit, and the handling Of the fruit at the packing
plant. Delay in any of these Operations contributes to
yield and/or quality losses.

Recommendations as to the best time Of harvest for
once-over harvested systems have been made (5, 6, 7, 8).
Some of these were purely intuitive while others were based
on conventional analyses given the existing data. Nearly
all of them recommend a time of harvest based on number
and/or color of oversized fruits (larger than 2 inches in
diameter) in the field. For example, Putnam (6) proposed
that the stage Of growth for maximum yield in once-over
harvesting of cucumbers usually occurred about 2 days follow-
ing the appearance of grade 3 fruit (1-1/2 to 2 inches in
diameter). Morrison and Ries (7) prOposed that the most
effective harvest index to Obtain the highest dollar yield

per acre was the development Of a yellow color on the most

mature fruit (2 to 2-1/2 inches in diameter). Miller and
Hughes (8) indicated that the maximum return per acre
occurred when the proportion of fruit greater than 2 inches
in diameter ranged from 14 to 31 per cent. The same source
also suggested that the planting should be harvested as
soon as fruit larger than 2 inches in diameter is found in
the field. The workers at the Asgrow seed company (5)
suggested that the maximum return in dollars will occur when
approximately 5 cucumbers per 15 feet of row (in twin-row
beds) have just begun to turn yellow at the blossom end.

Most of these harvest indices have the following
shortcomings:

(1) They are highly subjective in nature, therefore
they are hard to implement without the help of an
experienced person.

(2) The measure of their effectiveness is not clear
since, as indicated earlier, there is usually a
price contract between the processor and grower
which varies with the season and the location.

(3) They generally do not predict in advance Of the
harvest date.

(4) They do not predict product size composition, i.e.,
the distribution of number and weight of cucumbers

in the various size classes.

1.3 Objective

 

As an initial step toward eliminating these
comings, the following Objectives were selected for
present thesis:

(1) Development of a prediction model which can
to forecast 2-3 days in advance, the number
weight distribution of cucumbers in a field
prior to the normal time of harvest.

(2) Evaluation Of the above model for different

short-

the
be used
and the

just

loca-

tions in two growing seasons and for different

fruit price schedules.

(3) Devise possible methods to implement the above

model.

2. REVIEW OF LITERATURE

The factors that can affect the outcome of a
cucumber crop are numerous--type Of soil (5), fertilizers
(l, 9); irrigation (9), seed bed preparation, plant spacing
(5); plant population (9), planting schedules (5, 9);
weed control, insect control, bee activity, diseases
variety (5); climate (1, 9, 11); and other environmental
factors (10), etc. Of major interest in this study are
the factors that affect the rate of fruit enlargement at or
near the time Of harvest. The literature concerning this
aspect, particularly in the case of cucumbers, is rather
scarce. It is known, however, that daily fluctuation in
enlargement of fruits, in general, is a combination of
several factors, among which the intensity of the incident
solar radiation, the evaporating power of the air, and

night temperature play an important role (12).

2.1 Factors Affecting Rate of Fruit Growth

In the apple, enlargement of developing fruit has
been shown to occur mainly during the night (12, 13). The
rate of enlargement was a function largely Of night-
temperatures. Both under field and greenhouse conditions,

fruit enlarged only slightly during the daylight hours.

Further studies by Tukey (14) showed that the amount of
daily enlargement was influenced mainly by night tempera-
tures. A raising of night-temperature increased the amount
of apple fruit enlargement during the night, but reduced
the amount of daytime fruit enlargement. Went (15) found
that plants grown under a temperature cycle (diurnal
thermoperiodicity) of 78.8°F during the day-period and
64.4°F during the night-period grew vigorously and had more
fruiting compared to plants grown under a constant tempera-
ture. Aldrich and Work (16) reported that the rate of pear
fruit enlargement was influenced by soil moisture differences
within the available soil moisture range. Harley and
Masure (13) felt that with sufficient soil moisture, high
evaporating power Of the atmosphere had a greater retarding
effect on fruit growth than high temperature alone. Never-

theless, these two factors were found to be closely associated.

2.2 Factors Affecting Rate of
Growth of Cucumbers

 

 

The exact nature of the variables and the specific
mechanism'with which they affect the rate of fruit enlarge-
ment for cucumbers is largely beyond scientific under-
standing. Some workers believe, however, that the factors
which affect the rate of fruit enlargement for fruits in
general are also the factors affecting cucumber fruit
enlargement. The recent work Of Lewin (11) is a case in

point. Other workers have found that yield and cash

10

value per acre are more responsive to some factors than to
others. Among the most widely studied factors in the lit—
erature, under both categories, are: temperature, soil

moisture, plant population and response to nitrogen.

2.2.1 Temperature

 

Several workers have investigated the minimum and
optimum temperatures for growth of cucumbers. The work Of
Lewin (11) is of particular interest. He studied the
effect Of several controlled phototemperature and nycto-
temperature conditions on cucumber fruit enlargement
(expressed as fruit diameter enlargement) during a period
of 10 days after the fruit set, when grown under constant
irrigation and under a moisture stress condition. Some of
his findings were:

(1) The most favorable conditions for cucumber fruit
enlargement were a phototemperature Of 85°F and a
nyctotemperature of 65°F under constant irrigation,
or nyctotemperatures in the range of 55 to 75°F
under intermittent irrigation.

(2) Maximum daily enlargement generally occurred with
a phototemperature 20°F higher than the nyctotem-
perature.

(3) Cucumber fruit enlargement appeared to be limited
mainly by phototemperature, than by available soil
moisture and finally by nyctotemperature.

According to Bailey (17) the most suitable

11

temperatures for "rapid growth" were between 60 and 65°F at
night and up to 100°F in bright sunshine with an ample
supply Of soil moisture. Miller (1) reported that night
temperatures Of 60°F produced fruit with a greater length
to diameter ratio than those grown at 70°F. Studies by
Morrison (9) indicated that 50°F was approximately the
critical minimum temperature for develOpment of cucumber

plants and fruit.

2.2.2 Soil Moisture

 

Under controlled growth chamber conditions Lewin
et a1. (11) observed that the rate of cucumber fruit
enlargement appeared to be greater and more uniform when
the plants were not under moisture stress conditions for
approximately the first five days of enlargement after the
fruit set. During the second five day period, the moisture
conditions did not appear to influence fruit enlargement.

Morrison (9) Observed that inadequate soil moisture
resulted in poor seed germination, variable plant growth and
maturity. Workers at the Asgrow seed company (5) have
Observed that the most critical need for soil moisture for
cucumbers comes during the fruiting period, when lack of

moisture can seriously reduce the yield of marketable fruit.

2.2.3 Plant Population

 

Concurrent with the development of a machine to

harvest cucumbers in a once-over manner has been research

12

to study the most suitable plant populations. Putnam (6)
evaluated pOpulations ranging from 22,000 to 87,000 plants
per acre. P0pulations exceeding 43,560 plants per acre did
not produce higher yields.

Results from research conducted in North Carolina
indicated that the value of the crop increased with
increasing plant populations. However, at the highest
rates (224,000 plants per acre) the fruit, particularly the
larger sizes, tended to be pointed on the blossom end (18).

In studies with a once-over harvesting machine at
Michigan State University, the highest value in both bushels
and dollars per acre was obtained from plant populations of

31,500 in comparison to lower populations (19).

2.2.4 Fertilizers

 

Dearborn (20) found that high nitrogen increased
vegetative growth, fruit size number, and the percentage of
female flowers produced in cucumbers. He concluded that
a relatively high nitrogen level is essential for the produc-
tion of a maximum yield of highly-colored, straight, well-
shaped fruit. Vaile (21) reported that the set and number
of marketable cucumber fruit were greater under high than
under low nutrient conditions. Rodinkov (22) reported that
nitrogen was the most limiting nutrient during growth and
flowering, while potassium became the dominant element
during fruit formation.

Miller (1) found that 60 and 90 pounds per acre of

13

nitrogen reduced yields below those produced from 30 pounds
per acre. Morrison (9) observed that nitrogen rates of
approximately 60 pounds per acre were adequate for pro-
duction Of high once-over harvested yields. A recent
publication by McCOllum and Miller (23) indicated that
although the well-fertilized plants made more vigorous
growth than those with some nutritional stress, the increase

in the number of fruit set was only about one per plant.

2.3 Rate of Growth Curves for Cucurbit Fruits

 

Gustafson (24) constructed enlargement curves for
muskmelon, cucumber, scalloped summer squash and tomato
fruits. When the increase in fruit volume was plotted
against time, all fruits had the same general enlargement
curve: a slow increase in the beginning, followed by'a
very rapid increase, and then a gradual decrease and ces-
sation of enlargement. He compared the enlargement Of
fruits to that found for animals and vegetative plant struc-
tures which grow according tO the well-known "s" pattern.

Sinnot (25) divided the enlargement period of
cucurbit fruit into three parts: (i) from origin to
flowering, (ii) from flowering to the break in curve (the
end of exponential enlargements), and (iii) from the break
in the curve to final size. When the logarithms of fruit
volume were plotted against time, measured points fell

along an essentially straight line which flattened out at

the end

were as

(1)

(2)

(3)

(4)

(5)

14

Of the enlargement period. His main conclusions
follows:

Enlargement consists of two phases, an initial

one at a constant exponential rate, followed by a
phase at a continually decreasing rate until
enlargement ceases.

There is no relationship between enlargement rate,
either during the exponential period or later, and
final size.

There are differences in enlargement rates between
various genetic lines.

Environmental factors, insofar as these are reflec-
ted in seasonal differences, markedly affect the
enlargement rate.

There is an inverse relationship between enlargement
rate and enlargement duration in various seasons.
When the rate is high, final size is attained more

rapidly than when the rate is lower.

3. MODEL DEVELOPMENT

3.1 Existing Approaches

The harvest schedule of a crop grown for processing
should allow for the Optimum use Of all resources, including
equipment, labor and land. This necessitates a uniform and
continuous harvest sequence. To achieve this, a crop must
be planted in an orderly manner so that the quantity to be
harvested during any given period does not exceed the
harvesting or processing capacity (7).

Two methods have been suggested for timing succes-
sive plantings for mechanical harvesting. One method,
develOped by workers in Michigan and California (6), is
based on a stage-of—growth system, under which the second
planting is made when the first true leaves begin to emerge
between the cotyledons in the seedling of the earlier
planting. In areas where crops are seeded in midsummer for
late summer and fall harvest, this system Of timing may lead
to gaps and bunching at harvest time (6).

The other system, more widespread in use (26, 27)
is based on the accumulation of heat units (H.U.). This
system employs the use of a base temperature, below which
it is assumed no growth takes place. The mean Of daily

maximum and minimum temperature is determined, the base

15

l6

temperature is subtracted from this mean, and the resultant
value is referred to as heat units. For cucumbers, the base
temperature has been found to be 55°F and investigations (6)
have indicated that an accumulation of 75-100 H.U. is required
for germination and a total Of 850 to 1,000 H.U. is required
for the period from planting to harvest. In planning a
planting schedule according to the heat unit system, it is
important to consider such factors as the normal daily mean
temperature during the expected time of harvest and the

daily acreage capacity of the harvesting equipment (4).

The Michigan pickling cucumber industry utilizes
various heat unit systems to schedule crop planting such
that the harvest volumes are distributed over the harvesting
season-~approximately July 20 to September 15. However,
these systems are not sufficiently accurate for harvest
date scheduling. As pointed out by Arnold (28), heat units,
at least in their present state Of develOpment, are not
reliable because the relationship between temperature and the
rate Of plant develOpment is assumed to be linear when it is
undoubtedly curvilinear. He also suggested that factors in
addition to temperature affected the rate of development,
and that the temperature measured at a single location is
used as a basis for the prediction in the varied micro-
climate of many fields. Furthermore, a harvest date
scheduling system based on planting date, weather over the
growing season, soil type, etc. does not seem practical at

this time. The modeling work and the associated weather

17

monitoring system required would be very formidable

obstacles to such an endeavor (29).

3.2 An Alternative Approach

 

Some elementary notions from dynamical systems
theory suggested an alternative approach. In light of this
theory, we can consider our problem to be the description
Of the dynamics of fruit size distribution for a five—day
period just before harvest. We then define the universe
as everything outside this fruit size distribution. Inter-
action between these sets are abundant, such as the influ-
ence Of soil chemicals, moisture, light exposure, tempera-
ture, etc. on plant growth and hence on fruit growth.

Under controlled growth chamber conditions Lewin
et a1. (11) observed that maximum rate of growth Of cucumber
fruit occurred when the maximum air temperature was 85°F
and the diurnal cycle was 20°F. Physiological considera-
tions suggest that fruit development might be functionally
dependent upon both photosynthetic rate and moisture avail-
ability. Incident radiation was hypothesized to be indica-
tive of photosynthetic rate. Moisture availability is
critically dependent upon root development, soil type and
rainfall. Root development is in turn dependent upon soil
moisture conditions early in the growing season.

To include all Of the interactions such as described

above in a forecast model would necessitate a very complex

18

mathematical model. In general, the mathematical model
should be formulated to yield a reasonably accurate des-
cription or prediction of the behavior of a given system
while minimizing computational and programming cost and
time (43). Thus, boundaries were imposed on the scope Of
the defined problem. Direct measurement of factors such as
soil chemicals, nutrient levels, etc. were not included in
this study because they are difficult to quantify. The
effect of these factors, however, was studied indirectly
by estimating parameters from the Observed data for dif—
ferent locations (See Chapter 7). Measurement Of soil
moisture is not only expensive and time consuming, but this
particular variable can not be predicted accurately because
accurate rainfall forecasts are not available. Although
this factor is important, it was not possible to include
it in the prediction model. It was thought, however, that
some measure of its effect can be Obtained by estimating
parameters from the Observed data for different seasons.
Factors such as temperature, radiation and relative
humidity were included in the model because, not only is it
inexpensive to record them, but also their forecasts are
readily available.

Intuitively, it was then reasoned that if the state
of a field Of pickling cucumbers could be measured on day i,
then knowledge of input to the field (number of new fruits
and temperature were thought to be of major consequence)

might permit the prediction of the state Of the field on

19

day 1+1. Given this capability i can be set equal to 1+1
and the process can be repeated. To minimize the data
required for the parameterization of the prediction model,
only the time period coincidental with the 5 days just
prior to the optimum date of harvest was considered. This
limitation seems reasonable because a quick visual inspec-
tion is sufficient to determine whether or not a field has
entered this period. Furthermore, these inspections are a
part Of the current commercial mode Of Operation. Thus,
the result of these inspections can be utilized to initiate
the measurement of the system state.

The next step is the selection of the state variables.
The possible candidates for state variables in this problem
are the number and the weights of each fruit size grade. In
a later section we will establish that there are reliable
weight to number relations for fruits of various size
grades; therefore we need only consider fruit weight or
number as the state. It is helpful to consider here that
the output of the model in any case should be the weight of
cucumbers in various size grades. This is so because the
cucumbers are valued on the basis of the weight. This sug-
gests that, if a mathematical framework can be worked out
based on classification Of weight alone, we could select
weights as the state variables, which would then also be the
output of the model.

This possibility was, therefore, given consideration.

It was found early in the investigation, however, that the

20

selection of state variables based on weight, though pre-
senting no mathematical framework problems, does present
problems with regard to estimating parameters associated
with the growth of fruit. This notion is clearly demon-

strated by the following example. Let:

W

O weight of fruit of a given class c at

time T

zC = increase in weight of that fruit which remains
in class c during the time period T to T + AT

|-<
II

the weight of that fruit (measured at time T)
which was in class c at time T but in class c+1
at time T+AT
Y = the weight of that fruit (measured at time T+AT)
which was in class c-l at time T but in class c
at time T+AT.
Assuming no loss Of fruit from the system an input—output

relationship can be written as follows:

WC(T+AT) = WC + zc + Yc — Yc+l

The terms Zc' Yc and YC+1, however, can not be
measured from the Observed sample values and, hence, can not
be estimated. On the other hand, no such problem arises if
we consider the number rather than the weight of fruit of a
given class c.

Let:

N (T) = Number of the fruit Of a given class c at
c time T

Xc+1(T+AT)= the number of those fruit which were in class c
at time T but in class c+1 at time T+AT.

XC(T+AT)= the number Of fruit which were in class c-l at
time T but in class c at time T+AT.

21

Based on the assumption that there is no loss of
fruit and that there are two measurable states at time T
and T+AT, the following input-output relationship can be

written for the number of the fruit Of a given size grade c.
NC (T+AT) = NC(T) + XC(T+AT) - Xc+1(T+AT)

Notice here, the absence Of the variable ZC. This
particular elimination has been possible because a single
fruit during the period AT can not change its number (it
remains one); however, in the same period its weight does
change. Thus, the simplicity of the mathematical framework
dictated the use of numbers as the state variables.

The next problem considered was to relate the out-
put (weight) tO states (number). For this, it is helpful
to consider that cucumbers are divided into various size
grades based on diameter. Since diameters only help to
identify various classes, no single breakdown based on
diameter is unique. Any given breakdown, however, does
suggest the degree Of aggregation used in the model. This
implies that we can determine a classification, such that
meaningful functional characteristics exist not only for
state variables, but also between the state and the output
vafihbles. These functional considerations suggested a
classification of original number of fruit into six size
grades (see following section). Thus, while a classifica-
tion based on four size grades (sometimes used by the

industry) could result in both savings in sampling cost

22

and time, functional considerations challenged this approach.
The homogenity was somewhat better on the classification
based on six size grades. This classification resulted in
nearly constant weight to number relationships for the

six size grades.

3.3 Data Collection

 

In OOOperation with two Michigan pickle processors
some 200 once-over mechanical harvest fields were sampled
on successive days just prior to harvest in the 1971 and
1972 seasons. Moving diagonally across a field (typically
10-20 acres), 10 row lengths approximately equally spaced
on the diagonal were randomly selected. The row lengths
were selected such that a total of 100 square feet of area
was sampled (row lengths of 51-1/2 inches in 28 inch rows).
The plants in the selected row lengths were pulled, and all
of the fruit were removed, graded, counted and weighed.
Continuous recordings Of temperature, radiation and rela-
tive humidity were obtained in the area of the fields through-
out the harvest seasons. The data were collected from two
major pickle producing areas Of Michigan near Saginaw.
These areas were classified as West (Breckenridge and vicinity)
and North (Pinconning and vicinity). The exact location of
a field in each of the two areas was coded by a three digit
number. The number of successive daily samplings per field

was usually two to four. The last day of the sequence was

23

usually the day of harvest. Table 1 illustrates examples
of samples from two fields collected in 1971. Columns two
through seven show the number of fruit in each of the six
size grades (1A, 1B, 2, 3A, 3B and 4) as sampled in the
field on the corresponding date; columns eight through
thirteen show the corresponding weights. The size grades

based on minimum diameter measurements (Dmin) are:

Size 1A 3/8" < Dmin S 3/4"
Size 1B 3/4" < Dmin s 1-1/16"
Size 2* 1-1/16" < 0min s 1-1/2"
Size 3A 1-1/2" < Dmin 4 1-3/4"
Size 33 1—3/4" < 0min g 2"

Size 4 2" < Dmin

The other size grade classification occasionally
used by the processors to reduce the sampling labor and the
associated costs consists of dividing the sample into four
size grades rather than six size grades as shown above.

In this classification size 1A and 18 are lumped together
to form size 1. Similarly, size 3A and 3B are lumped
together to form size 3, the other sizes remaining unchanged.

The fruit size distribution shifts to increasing
numbers of fruit in the larger sizes as time progresses.
The cash value Of the field Often increases initially. For
most contract price systems, the value of size 4 (known as

F

*National Grader divides size 2 into 2A (1-1/16" <

< 1-5/16") and 28 (1-5/16” < Dmin é 1-1/2'9.

Dmin \

24

oversize or culls) fruit is zero. Typically, when a sub-
stantial number of fruit enters this size grade the value
of the field drops sharply. This can be seen clearly for

both fields illustrated in Table l.

3.4 Model Formulation

 

The Objectives of this study are stated in general
terms at the end of Chapter 1. Formulation of a mathemati-
cal model tO meet these Objectives involves consideration
of both the type of outputs desired and the type of inputs

available. These are as follows:

3.4.1 Available Inputs

 

(i) Number and weight distribution of fruit in a field
as sampled on a given day.
(ii) U. 8. Weather Bureau forecasts for the next three

days in the area of the field (temperature, etc.).

3.4.2 Desired Outputs

 

Number and weight distribution of fruit in the same

field for each of the next three days.

3.4.3 Model Structure

 

The structure upon which the proposed model character-
izes the dynamics of fruit number distribution is shown

schematically in Figure 1. As indicated in the schematic,

't“

25

 

musuosuum mmmaonuonEOc nonEdoso on» wo Emummflo ofiumsmnom 4

 

AB<+9O<H
Aevaaz

 

AB<+BVmH~
Aeomaz

 

Ae<+ev~7
Aevmz
Chi: 4m. N
Aevamz

ABQIJHmeZ Q Q AEGISmmx

Emmz

v
3.35 Q STE .x

 

Aevvz

“w

Ae<+ev.mx

19<+9V~x

“w

Ae<+evmax

.H ousoflm

e<+ev. x

26

the basic components of this structure are:
(1) number of fruit in the six size grades
(N4. N33, . . . , NlA) at time T
(2) number Of new fruit in the six size grades
(X4, X3B' . . . , XlA) at time T+AT
(3) number of fruit in the six size grades at time

T+AT.

The size (number) distribution of a particular class
of fruit at time T+AT is determined by their initial number
at time T and the number of new fruit at time T+AT. The
number Of new fruit of a particular size grade at time T+AT
is a function of initial distribution and the weather vari-
ables, Of which temperature is considered to be of major
importance. It is to be emphasized that the validity of
the resulting model as an instrument of simulation and pre-
diction depends critically on the degree to which time
behaviour Of number of new fruit can be characterized from
the available information and on the assumptions made in the
model structure. Basic to the structure proposed here is
the assumption that no fruit in any size grade is lost as

a result Of windfall.

3.4.4 Time Period

 

Mathematical models can be formulated in either the
continuous time form (described by differential equations)
or the discrete time form (described by difference equa-

tions). The choice of a continuous or a discrete time

27

model depends upon: (1) the level of detail necessary to
answer relevant questions; (2) the frequency Of events or
the flow rate Of Objects relative to the minimum time
interval of interest; and (3) the cost of programming and
Operating the models (30). The outputs from our model are
intended to provide information on a daily basis. The
input information is available on a daily basis only. The
consideration of these factors along with simplicity and
ease of computer implementation led to the selection Of a
discrete-time (stepping or recursive) model with a time
increment of one day. Thus, the AT in the ensuing discus-

sion refers to a time interval Of one day.

3.5 Functional Relationships

 

The flow of fruit (Figure 1) through various size

grades is characterized by the matrix S in the equation:

-> + +
NC(T+AT) = l NC(T) + §_XC(T+AT) 3.5.1
where:
F- — ‘1
N33 x33
—> _ N3A . i _ XBA
Nc _ ' c - x
N2 2
N13 X13
NlA xlA
L. __ a __

 

 

 

 

28

I is an identity matrix and S’is a (6 x 6) matrix

as shown below:

0 -1 1 o o 0
§_ = o o -1 1 o o
o o o -1 1 o

 

 

C. ._4

NC(T+AT) denotes the vector of the number of fruit
at time T+AT (e.g., NC(T+AT) represents the number of the
fruit of size grade c at time T+AT). NC(T) denotes the same
vector evaluated at time T. §C(T+AT) is a vector represent—
ing the number Of new fruit by size grade at time T+AT.

The notation §c(T+AT) is used rather than §C(AT) to
emphasize the fact that the observable states of the system
are at time T and T+AT. In effect, the number of new fruit
in each size grade are in transition, (Figure l), i.e., the
new fruit are in a size grade c-l at time T but they are in
the size grade c at time T+AT.

The matrices I and §_remain fixed from one period
to the next. The vector NC(T) represents the initial con-
ditions. Thus, knowledge of §c(T+AT) in 3.5.1 will permit
the prediction of NC(T+AT).

Since the sequences of field samples of count and
weight by size grade, together with the continuous record-

ings of temperature, radiation and relative humidity were

29

the major sources of data upon which a forecasting model of
new fruit could be based, the fruit development model for

new fruit was hypothesized to take the form:
+ + —>
XC(T+AT) = F[NC(T), W(AT)] 3.5.2

where W(AT) is a vector of variables summarizing the weather
occurring between sampling time T and T+AT.

Substituting 3.5.2 along with the initial condition
(NC(T)) in 3.5.1 will permit the computation of the state
at time T+AT(NC(T+AT)).

The next problem is to infer the relationships
between the output (weight distribution Of fruit of each
size grade) and the states. This problem will have a
relatively straightforward solution if it could be ascer-
tained that there is a definite relationship between the
number and the weight of fruit of each size grade c at a
given time T. Ideally, this would be the case if the
product under consideration is relatively homogeneous (a
fixed length to diameter ratio of fruit) and the number of
classifications of fruit size grade is infinite. This

implies,
WC(T) = e NC(T) 3.5.3

for each size grade c as the number Of size grades become

infinite. (WC(T) is the weight Of fruit size grade c.)

Let: —— —

2+
II

to

3’

 

 

Where:

+

WC(T) = a vector representing the weight of fruit at
time T by size grade. Thus, WC(T) is the weight
of fruit of size grade c at time T.

It!)
II

a (6 x 6) matrix of constants having non-zero
values on the diagonal and zero elsewhere.

The results of a preliminary regression analysis
with the six size grades over the period under considera-
tion (2-5 days before harvest) indicated that there are
fairly precise relations between the weight and the number
of fruit Of each size grade. The results of this analysis

are presented later (Section 3.7). Thus:

and
+ +
WC(T+AT) 2 E NC(T+AT) 3.5.5

That is, the distribution of the weight of fruit
(output) at time T+AT can be approximately determined from

the knowledge of the distribution of the number Of fruit

I111

 

 

V‘

A.‘

T 7'

‘Q

31

(states) at time T+AT.

From 3.5.1 we have:
-+ -+ +
NC(T+AT) — I NC(T) + §_XC(T+AT)
Premultiplying both sides of 3.5.1 by §_we have:

.+
g . I NC(T) + g . §_XC(T+AT)

_.)
g NC(T+AT)

or:

WC(T+AT) = g * _I_ NC(T) + g; * §Xc(T+AT) 3.5.6
Let:

E * I = P and E * S = P' 3.5.7
then:

H (T+AT ~ P N T + P' i (T+AT 3 5 8

c )- _ C() __ C > ..

Thus, the distribution of the weight of fruit at
time T+AT can also be determined from the knowledge Of the
distribution of the number of the fruit at time T and the
number Of the new fruit at time T+AT.

The matrices P and Rf can be determined from 3.5.7
or alternatively, they can be estimated directly from the
observed sample values. Let the estimated matrices corres-

ponding to g and 3' be Q and 9} respectively. Then,
-+ N + ' +
WC(T+AT) — Q NC(T) + Q_ XC(T+AT) 3.5.9

and:

32
+ +
P=Q;P'=Q'ifW(T)=ENc(T).
_ __ _ c _

That is, any improvement (in the sense of explain-
ing mean square deviation of the dependent variable) made
by the relation 3.5.9 over that of 3.5.8 will decrease our
confidence in the validity of our assumption in 3.5.4.

It is Obvious that the parameters in 3.5.8 can be
estimated by specifying the independent variables in several
combinations, of which 3.5.9 itself is a special case. An-
other form which is of considerable interest is as follows:

Let OC(T) represent the vector of number of old
fruit at time T by size grade, i.e., the number Of fruit
excluding the number Of new fruit in each size grade at
time T. Specifically:

04 (T) = N4 (T)

O3B(T) = N3B(T) - X4(T+AT)

O3A(T) = N3A(T) - X3B(T+AT)
3.5.10
02(T) = N2(T) - X3A(T+AT)
013(T) = NlB(T) - X2(T+AT)
01A(T) = N1A(T) - XlB(T+AT)
Substituting 3.5.10 in 3.5.1 we have:
+ + +
NC(T+AT) = I_oc(T) + I_XC(T+AT) » 3.5.11

Premultiplying 3.5.11 on both sides by §.we have:
+ + +
§_NC(T+AT) = §_oc(T) + g XC(T+AT)

then from 3.5.5 we have:

33
+ + +
We‘T+AT’ = §_OC(T) + §_XC(T+AT) 3.5.12

If instead, we estimate 3.5.12 directly from the

observed sample values we will have:
wcmwr) = g 5001‘) + g} §C(T+AT) 3.5.13

H and H' should be estimated to check the validity
of our assumption in 3.5.4 or 3.5.5. They can be employed
to determine the value of WC(T+AT)/OC(T) and WC(T+AT)/XC(T+AT)

if our assumption is not valid.

3.6 Parameter Estimation Procedure

The parameters in 3.5.2, 3.5.3, 3.5.9 and 3.5.13
were estimated by multiple regression utilizing the method
Of least squares (31, 32, 33). The general linear hypothe-

sis for R explanatory variables and N observations is:

Where:
Yt = Observation t Of the dependent variable Y
Xit = observation t Of the explanatory variable Xi
Ut = stochastic disturbance associated with observation t

34

2' so. bi no. bRo

and constants: b0, b1, b
The method of least squares consists of determining
estimates (b0, b1, b2 ... bi ... bR) of the constants b0,

b1! bz on. be

1 ... bR, such that the sum of the squared

residuals is a minimum, i.e.:
A

2
U

t
=1

”M2

is a minimum.

Where:

C)
O‘)

+ ... + ... + 8 x )

= Y - (b0 + b + b2X2t ixit R Rt

t t x1

1 t

A null hypothesis that the individual bi's equal
zero is established and tested to Obtain the regression
equation.

An important measure of how much Of the variation
in the dependent variable may be accounted for by the group
of explanatory variables is the coefficient of multiple
determination (R2). R2 is the proportion of the sum Of the
squared deviation from the mean of the dependent variable,
accounted for by the explanatory variables (32). The posi-

2

tive square root of R is the so-called coefficient of

multiple correlation.
3.6.1 Stepwise Addition Of Variables

In stepwise addition (34), a candidate (explanatory

variable) for entering into a least squares equation is

35

selected from among the independent variables not presently
in the equation. This candidate is the independent variable
which will reduce the unexplained sum of squared errors of

the dependent variable the most (equivalently, the candidate

2 the most).

is the independent variable which will raise R
If the candidate does not meet one of the stopping criteria
(a preset significance probability*), it is added to the
least squares equation, a new candidate is selected, and
the procedure is continued. If the candidate meets all
preset stopping criteria, the candidate is not entered into
the equation and the procedure is terminated.

There is the inherent danger in this procedure that
a group of variables which individually account for little
of the variation in the dependent variable, but as a group
explain much of this variation, may never be entered into

the equation (34). Therefore, a relatively high preset

significance probability level of 0.05 was used.

3.6.2 Stepwise Deletion of Variables

 

In stepwise deletion (35), an initial least squares
equation is Obtained using all of the independent variables.
One variable is then deleted from the equation and a new
least squares equation estimated. A second variable is

deleted and the least squares equation is recalculated. The

 

*Significance probability is the maximum probability
of rejecting the hypothesis: bi = 0, when bi = 0 (i.e.)
the probability of committing a type I error.

36

procedure continues until a variable selected as a candidate
for deletion meets one or more stopping criteria (a preset
significance probability). Since the selection of a candidate
variable for deletion is closely tied to the stopping criterion,
a preset significance probability level of 0.005 was used.

In general, the stepwise deletion involves moving
from a very general hypothesis (with many independent vari-
ables in the equation) to a progressively more restricted
hypothesis (with fewer independent variables in the equa-
tion). The stepwise addition moves from a very restricted

hypothesis to a progressively more general hypothesis.

3.7 Estimated Relationships

 

The estimation process combined statistical estimation
procedures and the author's intuitive understanding of the
cause-effect relationships governing the fruit development
process. All relationships were first estimated using step—
wise addition Of variables (Section 3.6.1) to allow for large
numbers of variables and to avoid the possibility Of singularity
problems. The method of stepwise deletion of variables (Sec—
tion 3.6.2) was also employed to permit all variable combi-
nations to account for the variation in the dependent variable.

The resulting regression relationships were then
scrutinized. Particular attention was given to the sign
and the magnitude of the coefficient Of each explanatory
variable, their standard errors of estimate as well as the

magnitude of the coefficient of determination (R2) and the

37

overall standard error of estimate for the relationship.
Least squares equations were then estimated (36) utilizing
the remaining explanatory variables. All relationships
were estimated from field samples of a loo-foot square
area. The notation used throughout is the same notation
formalized previously, with the following addition: R2, R,
and S.E. are the multiple coefficient of determination,
multiple coefficient Of correlation and standard error of
estimate, respectively. The values in parentheses directly
below the estimated coefficients will be the standard errors
of estimate of each of the coefficients.
The estimated relationships corresponding to 3.5.2
are as follows:
X4(T+l) = 30?36?3B(T) + 30?32?3A(T)
R2 = 0.50 R = 0.72 S.E. = 2 i~19M
X3B(T+l) = 36433N3A(T)
R2 = 0.48 R = 0.69 S.E. = 5 ..zd=b
"3A"““ ‘ ‘25?31§'2‘T’
R2= 0.32 R = 0.56 S.E. = 6

X2(T+1) = 0.25 N

(T)
(0.03)1B

II
\D
x

R2 = 0.20 R = 0.38 S.E.

38
XlB(T+l) = 0.30N (T) = 6
(0.07

R2 = 0.07 R 0.25 S.E. = 11 J‘-3

X (T+1)

II
0‘

S.E. = 15

In estimating the above relationships, an initial
hypothesis was made that fruit could not skip a size grade
from one day to the next. For example, fruit could not be
in size 2 today and size 33 tomorrow. The result of field
investigations on two experimental plots indicated that
this hypothesis is not true for the case of size 3A cucumbers.
A few size 3A cucumbers today may be size 4 cucumbers tomor-
row. Therefore, as indicated in the estimated relationship,
this possibility was permitted in the statistical estimation
process.

The constants in each of the first four estimated
relationships were omitted because of the following reasons.
(1) It is Obvious in these relationships that the dependent
variables must be zero if the independent variables are
zero. (2) A two—sided test of hypothesis (HO: O = 0 versus
H1: c f 0) established that the constants were not stat-
istically significant from zero at the 5 per cent signifi-
cance level.

The results Of field investigations as well as the
analysis Of the data indicated that the rate of formation

of new cucumbers per day for the size grade 1A and 13 can

39

not be explained by their initial distribution.

The climatic influence Of fruit development has not
been statistically verified. Extensive analyses of the
fruit development and the climatic data suggest that cer-
tain climatic variables may be statistically significant.
However, the variability in fruit develOpment explained by
them was very minor. The results of one such analysis,
based on the studies Of Lewin (11), are presented in
Appendix A. The exclusion Of measures Of soil moisture
stress is believed to be a major factor in these findings.
It is entirely plausible that in some of the sequences the
fruit develOpment was photosynthesis limited and in others
it was available moisture limited. If this is true the
influence Of temperatures and radiation would be expected to
be inconclusive.

The estimated relationships corresponding to 3.5.4
are as follows (all weights are in units of pounds):

W4(T) = 0.485 N4(T)
(0.005)

2

R = 0.982 R 0.991 S.E. 0.251

W (T) = 0.345 N (T)
33 (0.003)3B

2

I
C
o
w
0‘
b

1

b\

R = 0.982 R 00991 S.E. "

w (T)

N
3A 0.238 (T)

3A
(0.001)

R2 = 0.960 R

0.980 S.E. 0.406 .t‘

40

W2(T) = 0.127 N2(T)

(0.001)
R2 = 0.954 R = 0.977 S.E. = 0.375 r
WlB(T) = 0.043 NlB(T)
(0.0008)
R2 = 0.840 R = 0.917 S.E. = 0.249 1"

0.016 N1A(T)
(0.0004)

w1A (T)
R2 = 0.891 R = 0.944 S.E. = 0.056

The high values of the multiple coefficient of
determination (R2) and low values for the standard error
of estimate (S.E.) for the above six equations suggest
that there is a definite relationship between the weights
and numbers of a particular size grade.

It should be noted here that, in general, we do not
expect the relationship between weight and count Of 4 to be
constant with respect to time because the size grade 4 is
an absorbing state. However, the data employed in the
estimation were sequences that cover a relatively short
period Of time (usually two to five days before harvest).
It is quite plausible that during this period the relation—
Ship between the weight and the number of size four fruit
remains approximately constant.

The last two equations have slightly lower R2

values compared to the remaining four equations because of

(1) their greater variability in size, and (2) their

 

 

drg

41

greater range of diameter measurement. The estimated

relationships corresponding to 3.5.9 are as follows:

W4(T+l) = 0.482 N4(T) + 0.490 X4(T+l)
(0.010) (0.021)

R2 = 0.982 R = 0.990 S.E. = 0.253

W3B(T+l)

0.338 N3B(T) + 0.359 X3B(T+1) - 0.360 X4(T+l)
(0.009) (0.008) (0.025)

R2 = 0.983 R = 0.991 S.E. = 0.356

W3A(T+1) = 0.244 N3A(T) + 0.228 X3A(T+1) - 0.244 X3B(T+l)
(0.005) (0.006) (0.012)

R2 = 0.969 R = 0.984 S.E. = 0.392

W2(T+l) = 0.128 N2(T) + 0.120 x (T+1) - 0.126 X3A(T+l)
(0.002) (0.004) (0.007)

R2 = 0.956 R = 0.977 S.E. = 0.375

WlB(T+1) = 0.041 NIB(T) + 0.048 XlB(T+1) - 0.040 X2(T+l)
(0.003) (0.003) (0,004)

2

R = 0.856 R = 0.925 S.E. = 0.241

W1A(T+l) 0.016 N1A(T) + 0.016 X1A(T+1) - 0.016 XlB(T+1)

(0.0004) (0.0008) (0.0012)
R2 = 0.892 R = 0.944 S.E. = 0.057
The estimated relationships corresponding to 3.5.13

are as follows:

W4(T+1) = 0.481 x4(T+1) + 0.491 04(T)
(0.010) (0.021)

R2 = 0.981 R = 0.990 S.E. = 0.253

 

The

WBB (T+1) =

42

0.351 X3B(T+1) + 0.333 O3B(T)
(0.005)

R

0.982

‘w3A(T+1) = 0.230 X3A(T+1)
(0.004)

R2

0.967

W2(T+1) = 0.121 X2(T+1) +

R2

(0.004)

0.956

WlB(T+1) = 0.049 XlB(T+1)
(0.002)

R

= 0.855

W1A(T+l) = 0.016 X1A(T+l)
(0.0005)

R

= 0.892

The estimated matrices corresponding to E, g, Pf,

H and Hf referred to in the section on functional

ship are as follows:

~

 

0.485
0

0
0
0

0
0.34
0
0
0

(0.009)
R = 0.991 S.E. =
+ 0.246 O3A(T)
(0.005)
R = 0.983 S.E. =
0.128 02(T)
(0.001)
R = 0.977 S.E. =
+ 0.042 013(T)
(0.0009)
R = 0.925 S.E. =
(0.0004)
R = 0.944 S.E. =
0 0 0
5 0 0 0
0.238 0 0
0 0.127 0
0 0 0.043
0 0 0

0.361

0.399

0.372

0.239

0.057

2: .0."

relation-

COD

0
0.016

 

§=_E_, and

P'=

0.485
-0.345

 

(P—

0.482

 

0.480
-0.360

 

43

0
0.345
-0.238

0

0

0

0
0.338

0
0.359
-0.244

0

0

0

0

0
0.238
-0.127

0

0

0
0
0.244

0

0
0.228
-0.126

0

0

0.127
-0.043
0

0

0
0.120
-0.040

0

0 0
0 0
0 0
0 0
0.043 0
-0.016 0.016
0 0
0 0
0 0
0 0
0.041 0
0 0.016
.4
0 0
0 0
0 0
0 0
0.048 0
-0.016 0.016

 

 

 

 

 

0.491 0 0 0 0 0 fl
0 0.3 0 0 0 0
0 0 0.2 0 0 0
fl .-.
0 o 0 0.12 0 0
0 0 0 0 0.04 0
L— 0 0 0 0 0 0.016—
_ 0.481 0 0 0 0 0 —
0 0.351 0 0 o 0
and 0 0 0.230 0 0 0
H. = 0 0 0 0.121 0 0
0 0 0 0 0.049 0
0 0 0 0 0 0.016

 

On comparing the above matrices, we find that g = Q,

_1_>_' = g' (see 3.5.9), and also H = H' = I (see 3.5.13).
Furthermore, we note that the standard errors in the esti-
mated relationships of 3.5.4, 3.5.9 and 3.5.13 are nearly
the same. These results support the initial hypothesis of
3.5.4, i.e. , there is a definite relationship between the

number and the weight of fruits of a given size grade.

3.8 Estimated Models

The estimated relationships in 3.5.2 provide a model
to compute the distribution of new fruits at time (T+1) as

a function of the distribution Of the fruits at time T:

 

 

45

 

Sic(T+1) = g ficcr) + 9' i301) 3.8.1
Where:
_ 0 0.23 0.08 0 0 0 j
0 0 0.45 0 0 0
0 o 0 0.30 0 0
g = 0 0 0 0 0.25 0
0 0 0 0 0 0.30
0 0 0 0 0 0
._ o _
0
0
13' = 0
0
l -—(

 

 

U(T) =6 forT>,0.

It should be recognized that 3.8.1 is not a mathe-
matical relationship as we have shown above, but rather a
statistical relationship. To be precise 3.8.1 should be

written as follows :

xc(T+1) = g NC(T) + _B_' U(T) + t

where E is a vector of observed or estimated errors

 

 

 

 

Ur

 

46

corresponding to the assumption Of the error term in
ordinary least squares. We will, however, continue to dis-
regard it explicitly, but make note of it at appropriate
times in the discussion and interpretation of results.

Substituting 3.8.1 in 3.5.1 yields:

+

NC(T+1) = <1 + _S_C_) Nc(T) + 8.133601) 3.8.2

The above relationship indicates that the state of
the field at time T+l can be determined from the knowledge
of the state Of the field and the input at time T.

The weight of fruit in various size grades then,
can be determined either from 3.5.4 or 3.5.9 as follows:

+ +

WC(T) = §_NC(T) 3.5.4
or:

+ (T+1 - N (T + ' g T+1 3 5 9
Where: I, Q and g} are defined as above.

Equations 3.5.4 and 3.5.9 are strictly statistical

relations and not mathematical relations.

3.9 A Simulation Model

On the basis of the analysis carried out up to the
last section, it is easy to conclude that there is no one
unique way to present the model proposed; rather there are

several alternate, but equivalent ways in which the model

 

47

can be presented. One of these forms which is particularly
convenient because it has received wide attention in the
literature (37) is presented below.

Let:

NlA' NlB 3A 3B and N4 be the number of fruits

of size grade 1A, 1B, 2, 3A, 3B and 4 respectively in

a field sample Of area 100 ft.2.

, N2, N , N

WlA' W13, W2, W3A, W3B and W4 be the corresponding weight
of fruits in size grade 1A, 18, 2, 3A, 3B, and 4 res-
pectively.

T be the time in days, T = l, 2, 3, ...

M(T) be the maturity of a sample at time T, measured as

a fraction of number 2's in the sample, i.e.,

 

N2(T)
M(T) = _fi
N1A(T) + 1113??) + Nit-WT + N3A(T) + N3B(T) + N4 F‘T)"
Mi = Initial maturity of a sample as expressed above.

Mf final maturity of a sample as expressed above.
(Mi - Mf) represent the range Of maturity Of the sample

over which the model below is valid.

Then, for Mi 3 M(T) 5 Mf

NC(T+1) A ficm + 13 um 3.9.1

8 (‘1‘) = g; 8cm 3.9.2

 

.-.,

Where:

4.

“C (T)

|>

Iw

 

 

 

O

(“CO

48

N4 (T)
N3B(T)

N314 (T)

N2 (T)

NlB (T)

Nu. (T) .1

 

0.23
0.77

 

0.08
0.37
0.55 0.30
0 0.70
° 0
0 0
U(T) = 6
1 $ M(T) Mf
Mi 3 0.50
Mf 8 0.25

0.25
0.75
0

0
0.30
0.70

 

#

“I

an ‘

49

 

 

W4 (T)
W3B (T)
+ W3A(T)
WC (‘1‘) = W2 (T)
W13”)
E
‘3
r— -T
0.485 0 ° ' ' 0
0 0.345 0 ' ° °
' 0 0.238 0 ° '
E =
- ' ° 0 0.127 0 ‘
' ' ° 0 0.043 0
0 0 0 0 0 0.016
L_ ———.

 

 

Note that we have used maturity M(T) as a dununy
variable for time. Our model is valid in a period two to
four days before harvest. However, at time T=0, when the
first sample New) is taken, the day of harvest is not

known. Therefore, any reference to harvest time will not

be meaningful.

It should also be noted here that the exact relation-
ship between the time and maturity of a sample such as
described above is not known. However, the limits so and

25 per cent of number of 2's correspond approximately to

 

50

the time period of our interest. They also represent the

limiting sample values from which the model was estimated.

 

Appendix B contains the required matrices for a

simulation model based on size grades 1, 2, 3 and 4. It

is obvious that such a model is less desirable because of

the homogenity considerations discussed earlier in

Section 3.2.

4. MODEL VERIFICATION

Verification of the model was attempted using

both statistical tests as well as an examination of the

characteristic behavior of the system itself.

4.1 Statistical Tests

4.1.1 Model Structure

The model structure in 3.5.1 was based on (1)
availability of data (measurable states Of the system),

(2) theoretical considerations--fruits in a given size

grade during a given period can either enlarge to the next
size grade or remain in the same size grade, and (3) the
hypothesis that the rate of fruit develOpment depends upon

initial size distribution of the fruit at time T and

the
However,

the influence of weather during the period AT.

the influence of weather was not positively determined and

model in a relatively simple form such as the one

the
Recall that the parameters

given in 3.9.1 was proposed.

in 3.9.1 were arrived at, first by estimating 3.5.2 and
then by appropriate substitution Of 3.5.2 in 3.5.1. At

first one might be tempted to estimate 3.9.1 by ordinary

least squares directly from the Observed sample values.

51

52

The above reasoning can be made more precise as follows:

Since,
A (T+1) = I '15 (T) + s it (T+1) 3.5.1
C - c - C

and neglecting the effect of weather variables in 3.5.2

we have:

X4(T+1) = a NBB(T) + b N3A(T)

X3B(T+l) = c N3A(T)
X (T+1) = d N (T)
3" 2 4.1.1.1
X2(T+l) = e N1B(T)
X1A(T+l) = 9
Substituting 4.1.1.1 in 3.5.1 we have:
N4(T+1) = N4(T) + a N3B(T) + b N3A(T)
N3B(T+l) = (l-a) N3B(T) + (c-b) N3A(T)
N3A(T+l) = (l-c) N3A(T) + d N2(T) 4.1.1.2

N2(T+1) = (l-d) sz + e Nle

(T) + f N (T)

NlB(T+l) = (l—e) N 1A

1B
N1A(T+l) = (l-f) N1A(T) + 9

Then, estimating 3.9.1 directly corresponds to
estimating 4.1.1.2 from the Observed sample values. How-

ever, if the parameters a, b, c, , g are estimated using

the ordinary least squares, the estimated parameters a, b,

A

c, . . . , a will be biased and inconsistent. The reason

1

53

for this is that 4.1.1.2 contains not a set of six

independent equations, but a set Of six simultaneous
equations (See 3.5.1) . As a result, the errors and the

regressors (independent variables) are correlated. This,

however, violates the basic assumption of ordinary least

squares (32) that the error and the regressor are not cor-

"L

related. When this assumption is violated the ordinary

least squares results in parameter estimates that are

1" ‘W w ‘

biased and inconsistent (32, 38).

Methods are available for finding consistent and
unbiased estimates of parameters in simultaneous equation

models such as indirect least squares (32), instrumental

variable technique (32, 38), etc. It should be remarked

here that finding parameter estimates in 4.1.1.1 (as we
have done) rather than 4.1.1.2 using ordinary least squares

is a special case of the instrumental variable technique.

4 . l . 2 Test for Autocorrelation

Since the parameters in 3.9.1 and 3.9.2 were
estimated using time series data and also because non-

autocorrelation is generally assumed in ordinary least

squares, the residuals (estimated error terms) in 3.9.1

and 3.9.2 were examined for autocorrelation.

Autocorrelation is indicated, whenever the error

eT at time T is correlated with one or more of its previous

values (eT_1, eT_2, etc.). In ordinary least squares

estimation, the presence of autocorrelation signals

54

 

possible inadequacy of the regression model formulation.
Generally, autocorrelation does not destroy unbiasedness

and consistency of the estimates of the coefficients, but

rather of their variances (32) . When positive autocorrela-

tion is present, the variances of the coefficients are
generally underestimated leading to more frequent rejection
of the null hypothesis of hi equals zero.

A well-known test for the existence of autocorrela-
A test statistic d

 

tion is the Durbin-Watson test (39) .

for the null hypothesis (Ho: r = 0 versus H1 r > 0;

autocorrelation) of residual independence is computed.

1':
It

This statistic is also called the von Neuman ratio.
is the sum of squares Of the first differences of the least
squares estimated disturbances, divided by the sum of squares

Of the estimated disturbances, i.e.:

 

N
)3 c c 2
t=2 (Ut ' Ut-l)
d = N A
s at2
t=1

Where:
d = Durbin-Watson test statistic
Ut = the least squares estimator of the disturbance
for observation t.

If there is no autocorrelation, d is approximately

equal to two. Lower values Of d indicate positive corre-
lation, while higher d values indicate negative correlation.
(The regions of acceptance and rejection of the null

55

hypothesis are tabulated for comparison with the computed

d values (38).

Table 2 gives the computed values of d in 3.9.1

andEL9.2.

Table 2. Durbin-Watson test statistic (d).

 

 

 

 

 

L

 

—t3==$=============================
Relationship
(class) Estimated d

3.5.2 (4) 1.94
3.5.2 (3B) 2.06
3.5.2 (3A) 1.92
3.5.2 (2) 2.20
3.5.2 (13) 2.34
3.5.4 (4) 1.92
3.5.4 (3B) 1.67
3.5.4 (3A) 1.40
3.5.4 (2) 1.47
3.5.4 (1B) 1.85
3.5.4 (1A) 1.56

 

 

The critical values of d at d =

.01 level of

significance for testing the hypothesis H0: r = 0 versus

H

l
for K = 2 is 1.48; where K = number of regressors. On the

r >*() for a sample size of 60 and K = l is 1.45 and

basis of this test we cannot reject the null hypothesis

(H

O

3 r

0). Therefore, no autocorrelation is indicated.

 

 

56

The results Of the above tests in addition to the
one indicated before in the estimated 3.5.2 and 3.5.4 lead

us to believe that the estimated parameters in 3.9.1 and

3.9.2 are the best, linear, unbiased estimators of the

true population parameters.

4.2 Time Behavior

The response of the model to a given set of initial
conditions such as one Obtained by the standard sampling

procedure (See Section 3.3) is given in Figures 2 to 8.

These figures also show the Observed dynamic response of
The Figures

the system for the corresponding time period.

2 to 7 are drawn on the same scale to give an indication

of the relative magnitude of the different variables in-
All inputs and outputs are based on an area of

volved.
In each figure the ordinate for the number Of the

100 ftz.
fruit is given by the left hand Y-axis and the ordinate
for the weight of the fruit is given by the right hand
Y-axis. The abscissa in each figure represents the number
of days after the first sampling. All responses are shown

for a limited duration corresponding to the time period over
As indicated in an earlier section

which the model is valid.
this period corresponds approximately to the range of 50 to

25 per cent by weight of number 2'fruit in the sample.

An examination of the Figures 2 to 8 indicate that

the time behavior exhibited by the model is in close

 

 

57

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.N Thomas

 

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a
T.
..D
U.
3
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a
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0
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7.
mi
0H1

 

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1
O
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I
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M

I
o
‘1‘

 

Zn; 001 Jed 8.7 3° Jaqumu

58

.mm oomum Oman HON mosam> ooumasaam can Hmsuo¢

.m ousowm

 

VI;

Zn; 001 19d 8.8: go QQBIBM

 

 

mama
w m in
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on: \\ \
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on
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annuals III. I.

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.om

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rom

 

:3; 001 zed 8.8: 30 Jaqmnn

59

.em momuo ouam HOm mosam> ooumHszm can Hmouoa

.w madman

 

 

 

ammo
T 4 hi i? (H.

w 1 ‘\\\ .oa
m x
r. \\\
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m \

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60

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mama
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ooumaseflm

finance

 

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223 001 Jed 8.8 3° Jaqmnu

.om

 

 

61

Zn; 001 zed 3.8I 3° 445198

 

.mH oomum Oman “0m mosam> ooumHOEam ocm Hmsuoc

.o ousmflh

 

 

 

 

 

 

 

ammo
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c 1 VON
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3.

 

:33 001 Jed 8.81 :0 Jaqmnn

62

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.<H oomum mean How mosam> ooumaszm can Hmsuod .h ousoflm

<=’

mmmo
v m N H

b )P

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an; 001 16d s.v1 30 3H5T3M

0H1

 

 

omumasefim

Hmsuoa

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uoN

i.om

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Zn; 001 zed s.v1 3° zaqmnu

63

 

ucowms Hmuou pom
Hones: Hmuou now mosao> poundsfiam can amsuod .m ousofim

 

 

Zn; 001 zed atnz; :0 uqbtaM

mama
m N m a Tm
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an; 001 lad 3181; Jo Jaqmnn

64

agreement with the behavior Of the system itself. The dif-

ference between the actual and the simulated number of

fruits of a given size grade at a given time consists of

the following components: (1) sampling error, (2) measure—

ment error, and (3) stochastic error. Of these errors, only

the first two are controllable. Complete control over the

third type of error is not possible because Of the inherent ?‘

irreproducibility of the biological phenomena. This indi-

: “a“

cates that we do not expect the observed and the simulated
behavior to be exactly alike. However, we do expect that
the magnitude of the error relative to the magnitude of
the variable will be small. Figures 2 to 8 indicate that
this is true for all the variables under consideration.
Another important point which must be noted here is that
the errors in output (weight Of fruit) relative to the
states (number of fruit) are deamplified rather than ampli-
fied. This is not quite apparent from the figures, because
we have employed different scales for number and weight.
However, this can be seen to be true by examining the ele—
ments of the matrix I in 3.9.2. All elements in the matrix
I are found to be strictly less than one. The Figures 2 to 8
also suggest that the errors are not cumulative over suc-
cessive time periods.

It should be recognized that any inferences based
on only one sequence of Observations can be misleading.
This is so because our model is not deterministic, but

stochastic, because of the nature of the relationships in

65

3.8.1 which are statistical. Therefore, a more reliable
index of the quality Of the model can be Obtained by simu-
lating the model over all the sequences not used in
statistical estimation. The results of such a simulation
are presented in Table 3.

Table 3 indicates the mean and the standard devia-
tion of the percentage error of prediction of a given
variable over the remaining sequences (approximately 50%)
from West-1971. The definition of the mean percentage error
and its standard deviation for the variables in Table 3 are

as follows:

Let:
ANc(t) = observed number of fruit of size grade c
on day t
SN (t) = simulated number of fruit of size grade c
c
on day t
ENc(t) = percentage error in number of fruit Of size
grade c on day t
MENC = mean percentage error in number of fruit of
size grade c
SENC = standard deviation of percentage error in
number Of fruit of size grade c
AN(t) = Observed number Of fruit on day t
SN(t) = simulated number Of fruit on day t
EN(t) = percentage error in number Of fruit on day t

MEN = mean percentage error in number Of fruit

SEN = standard deviation Of percentage error in
number of fruit

c = size grades, c = 1A, 18, 2, 3A, 3B, 4

66

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.mm
b.0H We v.5 m4. 0.5 In ~30 NH m m e m s 0 IMm.
we.
_ 1
n.

w

«.0 0.~n 5.0: «.0 v.~ «.0 0.0 H H: o H N 0 H: w

3 v3 mmz £03 N3 mas daz z «z mmz mmz Nz maz «Hz
A unmflm3 ca muouuo ommucoouom Hogans ca muouuo oomucoouom
.m manna

.mmocosomm Ham «0 coach on» H0>o Houuo ommucmouom mo mcoauow>oo oumocmum can memo:

67

t = number of days in a given sequence, t = l, 2,

 

 

 

 

 

 

3' no.
n = total number of sequences used in simulation.
Then:
ANc(t) - SNc(t)
ENC(t) = x 100
AN(t) ._
r
n t (t+n)
N (t 2 t+ MEN 2
SEN - z ’3 E____.._° ) ( n) c
c " n t "’
(t+n-l) (t+n-l)
AN(t) = ANlA(t) + ANlB(t) + AN2(t) + AN3A(t) + AN3B(t) + AN4(t)
SN(t) = SN1A(t) + SNlB(t) + SN2(t) + SN3A(t) + SN3B(t) + SN4(t)
AN(t) - SN(t) 100
EN“) = AN(t) x
MEN = Z EN(t)
1'1 1'. (t+n)
and,
2 2
EN(t) _ (t+n)MEN
SEN = i E (t+n-l) (t+n-l)

The definition of the variables for weight is

similar to the one shown for number above and can be

68

obtained simply by replacing W for N at the appropriate
places. Table 3 further confirms our earlier Observations
Of the nature of the error term. As indicated by the
table, simulation errors in reproducing the past behavior
of the system are small.

The analysis in this section led us to believe
that our model is an adequate representation of the real
system. We note here that our model is valid only over a
limited time period corresponding to the range of 50 to
25 per cent by count Of number 2 fruit in the sample.
Fortunately, this is also the time period in which a
decision maker wishes to derive conclusions about the

system under consideration.

 

5 . SENSITIVITY ANALYSIS

The purpose of sensitivity analysis is to provide:

(1) greater insight about the inner workings of the simula-

'1

tion model, (2) an identification of the critical and less

critical parameters, (3) an indication whether some Of the

if“
i

constraints should be loosened or tightened, and (4) a
more quantitative idea about the expected overall perform-
ance of the system being modeled (40).

The parameters that can significantly influence
the performance of the model proposed are the parameters
estimated in 3.5.2. These parameters correspond to the
coefficients of the matrix Q in 3.8.1. Therefore, these
parameters were subjected to a sensitivity analysis (in
a very restrictive sense). These restrictions, however, are
due to the nature of the problem and not because Of the
computational cost. Any sensitivity analysis based on
variation in parameter value over the one given by the
coefficients of matrix g_is not meaningful here. This is
due to the fact that we know that the parameters in Q_are
the best, linear, unbiased estimates of the true parameters.
This was shown in the model validation section. It was
also shown in the same section that these parameters are

consistent estimates of the true parameter values.

69

70

A more meaningful index of the sensitivity of these
parameters can, however, be obtained by estimating the
matrix Q from the data collected from different locations
and in different seasons. This presumably will give us an
indication of the sensitivity of the parameters in g with
respect to both the location as well as expected changes
in growing season. The results of such an analysis for two
locations and two seasons are presented below. The loca-
tions in the present discussion will be referred to as West
and North and the seasons as 1971 and 1972 (See Section 3.3).
Thus, with the given information, it is possible to estimate
a set of four matrices, each corresponding to a different
location and season. Let these matrices be 91, 92, 93 and

94. Here,

C1 = estimated matrix from the data set West - 1971 = E
93 = estimated matrix from the data set North - 1971
C = estimated matrix from the data set West - 1972

C = estimated matrix from the data set North - 1972.

As indicated above, the matrix g_= 91 in 3.8.1 was
estimated from the data set West - 1971. The matrices £2,
93 and 94 were found as follows. The matrix Q_= g1 has
been repeated for the sake Of comparison.

71

 

 

 

 

 

West - 1971
0 .23 .08
0 0 .45
0 0 0
E3-1 = 0 0 0
0 0 0
0 0 0
North - 1971
0 23 .04
0 0 .46
0 0 0
C =
-2 0 o 0
0 0 0
0 0 0
West - 1972
0 .32 0
0 .48
0 0

O
0000
O

 

 

 

 

72

North - 1972

 

 

F— O .15 .10 0 0
0 0 .44 0 0
0 0 0 .30 0
94 =
0 0 .0 0 .25
0 0 0 0 0
__ __J

 

 

g3 and 94 are of dimension (5 x 5) because the 1972
data did not include the decomposition of size 1 fruit into
sizes 1A and 13. When this decomposition is not done,
samplers tended not to include the fruit of size 1A into
their samples because the fruit of size grade 1A are very
small in diameter and length. Therefore, the size grade 1
in 93 and 94 should be interpreted not as l but rather 1B.

Comparing 91 with 99, 93 and 94 leads to the follow-
ing general conclusions:

(1) The distribution of new fruits xc(T+l) as a func-
tion of the distribution of the Old fruits NC(T)

in a given season does not appreciably change

with the location.

(2) For a given location, the seasons do not appre—
ciably affect the distribution Of new fruits

XC(T+1) as a function of the distribution of the

old fruit NC(T).

It should be noted here that the above generalizations

 

 

Te

Tl

 

73

do not apply to fruits of size grade four. Also, inherent
is the assumption that in making these generalizations the
cultural practices remain the same within different loca-
tions and seasons.
Further investigations into the behavior of fruits
Of size grade four led to the following conclusions:
(1) The reason that the estimated parameters for the

new fruits of size grade four X4(T+l) as a func-

ilrliw'w' ‘m— ‘ I _~—'
‘\

tion of the old fruits Of size grade 3B and 3A
[N3B(T) and N3A(T)] vary among locations and
seasons, is that the populations from which the
samples were drawn were at different average
maturity levels* (See Table 4).

Table 4. Mean and standard deviation of fruit of size
grade four for different locations and seasons.

 

 

 

 

 

Standard

Location Season Mean Deviation
West 1971 4 3
North 1971 4 3
West 1972 6 5
North 1972 3 2

 

 

 

*Measured as average number of fruit of size
grade 4 in the data set of a given location and season.

 

74

(2) A higher average maturity level was found to be
associated with an increase in the magnitude Of
the parameter C12 (coefficient in the first row
and second column of the matrix Q) and a decrease
in the magnitude of the parameter C13.

(3) A lower average maturity level was associated with
a decrease in the magnitude of the parameter C12
and an increase in the magnitude of the parameter
C13.

The above findings are consistent both in theory
as well as in actual field experimentations. It can be
argued that at relatively low maturity levels where the
growth of fruit is not limited by the available nutrient
and moisture supplies, it is possible that comparatively
more fruits which are size grade 3A today grow to size
grade 4 tomorrow. However, at a higher maturity level,
where the growth is limited both by the available food
supply and the competition among fruits of various size
grades, we would expect the growth of the fruit of size
grade 3B to size grade 4 to be much higher than the growth
of fruit Of size grade 3A.

The results of actual field experimentations pro—
vided further support to the above theory. In these experi-
ments actual observations were made on a given number of
fruits Of various size grades over a relatively long dura-
tion Of time. All fruits in a selected area of 50 ft2 were

labeled. Diameter measurements on these fruits were made

75

daily for 9 days, the harvest date being approximately in
the middle of this nine-day period. From these diameter
measurements the number and the weights Of fruits of
various size grades were calculated for each day. The
results of an earlier study (41) indicated that the weight

is related to the diameter by the following relationship:

1.96D
W = 3.8 e

Where:

W = weight of fruit in grams
and:

D = diameter of fruit in inches.

The results Of one such diameter study for a
limited duration are given in Table 5. An analysis of
transition behavior of fruit in various size grades from
these diameter studies suggested that the conclusions
numbered one through three are reasonable. Further experi-
mentation Of a controlled nature is needed to get more
positive results.

The discussion so far has addressed itself to a
broad aspect of the problem, identifying the sensitive
parameters (coefficients C12 and C13 in the matrix Q)
and the relative significance Of these parameters with
respect to time in the model. TO use the above information
for making decisions, however, one requires reliable
estimates of the numerical values of these parameters.

This latter aspect of the problem for the given

 

76

model can be answered best by means of a parametric
simulation. For our purpose, we will define parametric
simulation as a process in which the only change from one
simulation run to the next is in the level of a given
parameter or parameters. The initial conditions and the
input remains the same in all simulation runs.

The results of one such parametric simulation at
four levels Of C12 are presented in Table 5. These levels
are meaningful intuitively, but otherwise are arbitrary.
The parameter C13 was retained at its original level and
was not subjected to such a study because its contribution
in explaining the behavior of fruit Of size grade four is
very small. Table 5 also shows the actual ObserVed dynamic
response of the system for the given time period as deter-
mined by the diameter study.

The results Of Table 5 indicate that at low maturity
levels (first three days), the simulation results remain
unaffected for any parameter value within the range of 0.23
to 0.55. However, the effect begins to become apparent on
the fourth day. At high maturity levels the growth of
fruit of size grade four increases dramatically with the
increase in magnitude of parameter C12. This implies that
any parameter C12 such as one found in matrices 91' 92, 93
and 94 will be adequate if the Object Of the simulation is
to predict the distribution of the fruit of various size
grade at low maturity levels (approximately 50 to 25 per

cent by count Of number 2 fruit in the sample) only.

77

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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78

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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79

However, the usefulness of the model might be extended to
include high maturity levels (50 to 15 per cent of number 2)
simply by using a relatively high magnitude of the parameter

C12 such as 0.45 to 0.55.

6. LINEAR DYNAMIC MODELS

The model presented in 3.9.1 and 3.9.2 is a class
Of models known as linear dynamic models. These are des-

cribed (37) as follows:

Let:
+
U(T) = input or stimulus vector which includes the
m inputs
+
Q(T) = state vector that includes the n "internal"
variables which relate the input to the output
and which account for past inputs and states
and
+
Y(T) = output vector that includes the q outputs Of the

system model.
Usually m.s n and q s n. Otherwise, some of the
inputs and/or outputs are redundant. Using the above
notation the structure of a linear constant-parameter

discrete time state model is given as follows:

5(T+1)

z_s._ 6m + 2 601) 6.1

Y(T) = 601) 6.2

It!)

Comparing 3.9.1 and 3.9.2 with 6.1 and 6.2 we

80

81
We (T) = Y(T)

Thus, it is seen that our model is a linear dif-
ference equation model. The theory for such models is well
developed (37) and they can be shown very conveniently in
the form of a block diagram such as the one shown in Figure
9. The overall system behavior of such models can be
determined either by a direct simulation Of the system from
the block diagram or from the formal solution Of the ag-
gregated model.‘ Of course, either solution yields the

same input-output behavior.

6.1 Analytic Solution

 

Given the initial condition Nc(0) and the input
U(T), T = 0, l, 2, ... t-l, the general solution NC(T) for
3.9.1 can simply be found by iterating for T = 0, 1, 2 ...

etc. Thus, at T = 1

-> + +
Nc(l) = §,NC(0) + E U(O)
at T = 2
+ 2 + +
Nc(2) = A_ NC(0) + §_§_U(0) + B U(l)
at T = 3
+ - 3 2 + +
Nc(3) — A NC(0) + A B U(O) + 580(1) + B U(2)

I

82

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.0 ouson

 

 

 

 

 

 

SEA m Em .m. + 2.6.0 n 3130.
:00

 

 

 

 

 

 

3.0m

83
New) = At New) + At'l g I? (0) + if"? g 6 (1) +

+ 1 piflt-l)

or:
New) = _t New) + of“, 5V2, at”, , _1_1 20(0)
+
8 0(1)
+
13 0(2)
— B U(t-1)__j

 

 

and in summation form:

t
New) = 5" N’ (0) + z
1

The stability behavior of the states is determined
by At which will grow or decay with increasing t depending
upon the eigen-values of A, If they all are less than
unity, the system is asymptotically stable (37). Based on
this criteria our model in 3.9.1 is found to be unstable.
However, we note here that the model in 3.9.1 is developed
only to study the transient rather than the steady-state
characteristics of the system. Due to the instability,
steady-state conditions do not exist. This is so because
we are simulating a complex process via a simplified model.
It is for this reason that the model was claimed to be

valid only over a limited range of field maturity.

7. CASH VALUE PREDICTION

The Objectives of this study were stated in general
terms at the end of Chapter 1. One of these Objectives
was to measure the performance of the model in predicting
the cash value of fruit from a given production unit. In
this chapter the quality of forecast of cash value by the
model is evaluated.

The value of a cucumber crop is assessed on a
weight basis. A survey Of the industry indicated that both
aggregate as well as differential pricing systems are in
vogue. In either case, the value of the product of size
grade four at any time is zero. The following examples

give an indication of the two pricing structures.

7.1 Aggregate Price Structure

In aggregate pricing the value of the product is
assessed on the useable weight basis. Useable weight
[UW(T)] at any time T is defined as the total weight
[W(T)] of the product minus the weight of the product of

size grade four [W4(T)]. Thus:
UW(T) = W(T) - W4 (T)
Then, the value of the crop per acre at a given

84

85

time T is given as follows:

A(T) = a°b-[W1A(T) + wlB(T) + W2(T) + W3A(T) + W3B(T)]
or:

A(T) = a-b-UW(T)
Where:

A = value of the crop, $/acre

a = a constant depending upon the area used in the
sampling procedure. For the standard sampling
procedure described in this study a = 435.0. This
implies that 100 ft2 area is covered in obtaining
the sample.

b = a constant with units of dollars/pound. Its par-
ticular value is agreed upon in a contract between
the producer and the processor.

WC = refers to the weight of fruit of size grade c,
c = 1A, 18, 2, 3A and 3B in a given sample.

7.2 Differential Price Structure

 

In differential pricing the value of the produce is
assessed on the basis of the individual weights of fruit of

various size grades in the sample.

e3A'W3A(T) = e3B'W3B(T) + e4-w4('r)]

Where:
D = value of the crop, dollars/acre

a = 435.0

86

.uoH>msmn 09Hm> ammo oODOOme

wasp cH TEHB

. 0H 033m

 

 

(8109/5) ezov Jed anteA qsso

87

e0 = value of the product of size grade c, c = 1A, 1B,
... , 4 in dollars/pound. Usually e1A 9'918 > e2 >
e3A > e3B > e4. Typically e4 = 0.

The two pricing structures discussed above may
have advantages and disadvantages over one another. This,
however, is not the basic issue here. The issues to be
addressed here are:

(1) Given a price schedule, how can it be used in
making decisions regarding scheduling of once-
over harvest?

(2) To Obtain some quantitative measure of the per-
formance of the model in predicting cash value of
the fruit based on the above pricing schedules.
The answer to the first question can be given by

the theory and an understanding Of the system's behavior.
Thus, if cash value per acre (obtained from either schedule)
is plotted against time, its expected behavior will be as
shown in Figure 10. The figure shows a steady increase in
cash value reaching a peak and then a steady decline as
time progresses. This particular behavior in aggregate
pricing is expected because of the zero value of fruit of
size grade four and in differential pricing this feature
combined with the relative weight factors ($/lb) of fruit
of different size grades. Thus, it is Obvious that any

of the price schedules can be used for locating an Optimum
day of harvest.

Figure 11 shows the behavior of useable weight

88

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mama
h 0 m
P F

b

.HH magmas

b

D

1m

r0H

IMH

10N

1mm

 

(spunod) quraM atqeesn

89

over a relatively long time period (9 days) determined
from two diameter field studies referred to in Chapter 5.
Figure 11 underscores the fact that the useable weight and
hence, the cash value over time for some fields, may have
a more pronounced peak than for other fields. In such
situations a decision as to the time Of harvest should be
based not only on the consideration Of Optimal cash value,
but also the product size composition.

To deal with the second issue a number Of simula-
tions were conducted on data—sets from all locations and
all seasons. The simulation results relate to not only
the narrow problem Of cash value prediction, but also gives
an indication of the overall performance of the model.
Simulations were conducted to determine the model per-
formance with respect tO (l) one-day forecasts, (2) two-
day forecasts, and (3) up to five-day forecasts. The
third gives an indication of the expected errors in fore-
casts of greater than two days. The results of these
simulations are presented in Tables 6 through 10. The
notation used in these tables is the same as given in the
section on model validation. In all cases considered the
standard deviation of error in predicting cash value is

approximately 15 per cent.

90

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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91

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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8. USE OF THE MODEL

Whenever a complex system is modeled, the behavior
of the system predicted by the model must be reviewed
critically. All of the assumptions used in developing the
model must be considered; all of the possible ways in which
the model might differ from the real system must be examined.
The behavior predicted for the system can only be as accu-
rate as the model used to represent the system. Therefore,
the modeling of a complex system should generally be regarded
as an aid to decision making rather than as a decision making
process itself (42).

In specifying the way in which the model results
relate to and can be used in actual management practice,
several items must be considered. The fruit distribution
or the cash value suggested by the model for a given field
on a given day represents not the distribution or the cash
value that one could verify with one sample by the procedure
described in Section 3.3, but is rather the expected dis-
tribution or the cash value that one would find through
the repeated application of the sampling procedure.

The model has been verified only for those varieties
and cultural practices common to Michigan commercial pickling

cucumber production. Extremes of weather (e.g., cold

95

96

shock) are not adequately considered in the model either.

The assumption was made in developing the fruit
number dynamics structure that no fruit is lost as a result
of wind or other natural causes in transition from one size
grade to the next. This assumption should be further veri-
fied. It appears from the author's own observations of
field plots that this assumption is not critical in making
a short range (e.g., 3 days) forecast, but its accumulated
effect in long range (e.g., 9 days) forecasts needs further
investigation.

It should be apparent from the above discussion
that although the results from the model presented can be
used as a guide in determining fruit size distributions
and/or cash values, the role of the manager in the actual
decision-making process is of critical importance in
achieving a successful transition between the model and

reality.

9 . IMPLEMENTATION

Two methods of implementation of the fruit develop-
ment model for forecasting the optimum day of harvest were
considered. One possibility is to summarize the major
features of the model in tables or charts to be used in
the field immediately after a field sample is taken.

(Two different forms have been developed and subjected

to limited field tests.) The implementation costs (time)
and dollars) are very low but the information obtained may
not be adequate.

Most experience to date, however, is in the utili-
zation of a teletype-telephone hook-up to a commercial time
share computing system. Such a system is surprisingly
inexpensive when utilized for all fields of cucumbers con-
tracted by a processor. Total fixed hardware costs (tele-
type rental and supplies) are less than $200 per harvest
season. Individual field forecasts are then obtained at
a cost on the order of $0.50 to $1.00 (telephone and com-
puter charges). In addition to the forecast itself an
accounting capability is readily available which could be
used to chronologically order all of the expected harvests
for a processor. Thus, a total daily fruit volume flow into

the plant forecast would also be available.

97

98

Dr. J. B. Holtman, major professor of the author,
in collaboration with Mr. Robert Milligan of the Agricultural
Economics Department of Michigan State University, has
developed a program for the implementation of the model for
a small commercial time-sharing system. This program
stores the incoming field samples and returns a forecast
of fruit size distribution and cash value for each of the
next three days. Utilizing the teletype-telephone hook-up
to the time shared system, two Michigan processors evaluated
the system during the 1971 season. It was found that this
system was a highly reliable and efficient method of imple-
menting the forecasting system. After only four hours of
training, a plant clerk (who previously had no related

experience) was utilizing the system with no difficulty.

10. SUMMARY AND CONCLUSIONS

The general problem of scheduling formed the basis
of the thesis. In particular a cucumber fruit development
model was developed to assist the manager in making a
harvest time decision for once-over mechanical harvest.

The application of the model for the grower is in making
reliable forecasts of the harvest date of a field 2-3 days
in advance to optimize his expected gains. The application
of the model for the processor includes scheduling to reduce
field losses due to unforseen shortages of equipment and/or
labor and regulation of the flow of desired size distribu-
tion mix into the plant.

The model proposed is a class of general linear
dynamic models of discrete time and constant parameter form.
The model can be viewed in light of discrete state space
theory or population dynamics. Both views are helpful.

The states of the model are the number of fruit of various
size grades at time T. The output from the model is the
weight of fruit of various size grades at time T. The

input to the model is a unit step that corresponds to a
constant number of new fruit of smallest size grade entering
into the system each day. The parameters in the model were

estimated using the technique of multiple regression with

99

100

least squares criterion.

Model verification was attempted using both the
available statistical tests as well as analyses of the
time behavior from the known historical record. Model
verification along with sensitivity analysis has proved the
general adequacy of the model in making harvest time deci-
sions for different locations and seasons.

The model was implemented for on-line applications
via a teletype-telephone-time share computer hook-up.

The following general conclusions can be made:

(1) The model proposed for cucumber fruit development
is valid under the assumptions made in its
development.

(2) The model shows promise of being a valuable tool
in accurately scheduling 2-3 days in advance the
optimal harvest date.

(3) The model implementation is feasible economically

and operationally.

(1)

(2)

(3)

(4)

(5)

ll . RECOMMENDATIONS

Further refinement in the model can be accomplished
by studying the effect of environmental variables
on the system considered. Basic studies are needed
to relate the effect of such variables as tempera-
ture and moisture to the rate of development of

new fruits.

The exact behavior of the fruit of the smallest
size grade (input) needs to be further explored.
The assumption that the loss of fruits as a result
of wind is negligible, needs experimental verifi-
cation.

The broad framework presented in this thesis for
fruit population dynamics is applicable and use-
ful for other types of population studies where
interactions depend upon population member size as

well as time.

A detailed study of variations among varieties

should be considered.

101

 

 

REFERENCES

10.

REFERENCES

Miller, C. H. 1957. Studies on the nutrition and
physiology of pickling cucumbers. Ph.D. Thesis,
Michigan State University.

Research project report on post-harvest handling of
cucumbers. 1970-72. Department of Agricultural
Engineering, Michigan State University.

Stout, B. A., M. M. DeLong, D. H. Pettengil, and
S. K. Ries. 1964. A once-over mechanical har—
vester for pickling cucumbers. Mich. Agr. Exp.
Sta. Quart. Bul. 46(3):420-430.

Sims, W. L., M. B. Zahara. 1968. Growing pickling
cucumbers for mechanical harvesting. Agri. Ext.
Ser. University of California, AXT-270.

Growing cucumbers for processing. 1968. Asgrow Seed
Company, Orange, Connecticut.

Putnam, Alan R. 1963. Horticultural aspects concerned
with the production of pickling cucumbers for once-
over harvest. Thesis for the degree of M.S.,
Michigan State University.

Morrison, F. D. and S. K. Ries. 1967. Cultural
requirements for once-over mechanical harvest of
cucumbers for pickling. Amer. Soc. Hort. Sci.
91:339-346. '

Miller, C. H. and G. R. Hughes. 1969. Harvest indices
for pickling cucumbers in once-over harvested
systems. Amer. Soc. Hort. Sci. 94:485-487.

Morrison, F. D. 1966. Cultural and environmental
parameters for mechanically harvested cucumbers.
Ph.D. Thesis, Michigan State University.

Hopen, H. J. 1962. Environmental factors affecting

the growth of guggmifi,53gign§ L. with special
reference to carbon dioxide. Ph.D. Thesis, Michi-

gan State University.

102

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

103

Lewin, I. J. 1970. Effect of photoperiod and nycto-
period temperatures, and moisture stress on fruit
enlargement in the cucumber, Cucumis Sativus L.,
CV. Mini-Cuke. Ph.D. Thesis, The Pennsylvania
State University.

Tukey, L. D. 1962. Factors affecting rhythmic diurnal
enlargement and contraction in fruits of the
apple (Malus domestica Bork.).

Harley, C. P. and M. P. Masure. 1938. Relation of
atmospheric conditions to enlargement rate and

periodicity of Winesap apples. J. Agri. Res.
57:109-124.

Tukey, L. D. 1960. Some effects of night temperature
on the growth of McIntosh apples. II. Proc. Amer.
Soc. Hort. Sci. 75:39-46.

Went, F. W. 1944. Plant growth under controlled
conditions, II. Thermoperiodicity in growth and
fruiting of the tomato. Amer. J. Bot. 31:135-150.

Aldrich, W. W. and A. Work. 1934. Evaporating power
of the air and top-root ratio in relation to rate
of pear fruit enlargement. Proc. Amer. Soc. Hort.
Sci. 32:115-123.

Bailey, L. H. 1947. The Standard Encyclopedia of
Horticulture. 905-907. The MacMillan Company,
New York.

Bailey, L. H. 1963. Unpublished data.

Bailey, L. H. and A. R. Putnam. 1963. The feasibility
of a once-over mechanical harvester for pickling
cucumbers. Mich. Agr. Exp. Sta. Quart. Bul.
42:2-23.

Dearborn, R. B. 1936. Nitrogen nutrition and chemi-
cal composition in relation to growth and fruiting
of the cucumber plant. Cornell Univ. Agri. Exp.
Sta. Memoir 192.

Vaile, J. E. 1942. Ark. Expt. Sta. 53rd Ann. Rpt.
Bul. 417:28.

Rodnikov, N. I. 1944. The rate of maturity and the
yield of cucumbers grown under glass and their
relationship to mineral nutrition. Proc. Sci.
Conf. Timirjazev Agri. Acad. 3-10 June, No. 1,
pp. 45-46 (Hort. Abst. 16:39).

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

104

McCollum, R. E. and C. H. Miller. 1971. Yield,
nutrient uptake, and nutrient removal by pickling
cucumbers. J. Amer. Soc. Hort. Sci. 96(1):
42-45.

Gustafson, F. G. 1926. Growth studies on fruits.
Plant Phys. 1:265-272.

Sinnot, W. E. 1945. The relation of growth to size
in cucurbit fruits. Amer. J. Bot. 32:439-446.

Bomalski, H. H. 1948. Growing degree days. Food
Packer, 29:51-59.

Reath, A. N. and S. H. Wittwer. 1952. The effects
of temperature and photoperiod on the development
of pea varieties. Proc. Amer. Soc. Hort. Sci.
60:301-310.

Arnold, C. Y. 1959. The determination and signifi-
cance of the base temperature in a linear heat
unit system. Proc. Amer. Soc. Hort. Sci. 74:
430-445.

Holtman, J. B., A. Patel, F. Panol and B. F. Cargill.
1972. A mathematical model to schedule cucumber
harvest. Amer. Soc. Agri. Eng. 72-149.

Manetsch, T. J. 1971. Unpublished class notes
prepared for systems science 812. Sys. Sci.
Dept., Michigan State University.

Kmenta, J. 1971. Elements of Econometrics.
MacMillan, New York, 750 pp.

Kane, E. J. 1968. Economic Statistics and Econo-
metrics. Harper & Row Publishers, New York,
437 pp.

Draper, N. R. and H. Smith. 1966. Applied Regres-
sion Analysis. John Wiley 8 Sons, Inc., New
York, 407 pp.

Rafter, M. E. and W. L. Ruble. 1969. Stepwise
addition of variables. Agr. Expt. Sta. Stat.
Series Description No. 9. Michigan State Univ.

Rafter, M. E. and W. L. Ruble. 1969. Stepwise
Deletion of Variables. Agr. Expt. Sta. Stat.
Series Description No. 8. Michigan State Uni-
versity.

36.

37.

38.

39.

40.

41.

42.

43.

105

Rafter, M. E. and W. L. Ruble. 1969. Calculation of
Least Squares Problems. Agri. Expt. Sta. Stat.
Series Description No. 7. Michigan State Univ.

Manetsch, T. J. and G. L. Park. 1972. System
Analysis and Simulation with Applications to
Economic and Social Systems. Preliminary Edi-
tion. Department of Electrical Engineering and
Systems Science. Michigan State University.

Wonnacott, R. J. and T. H. Wonnacott. 1970.
Econometrics. John Wiley 8 Sons, Inc., New York.

Durbin, J. 1960. Estimation of parameters in time
series regression models. Journal of the Royal
Statistical Society, Vol. 22, Series B, pp. 139-
153.

Asimow, Morris. 1962. Introduction to design.
Prentice-Hall Inc., Englewood Cliffs, N. J., 135pp.

Bingley, G. W. 1959. Construction, evaluation, and
efficiency studies of a mechanical cucumber
harvester. M.S. Thesis, Michigan State University.

Patel, Ashok K. 1970. Development of a forage-feed
planning model for the Rutgers dairy herd: An
activity analysis approach. M.S. Thesis, Rutgers
University, N. J.

Naylor, Thomas H., Joseph L. Balintfly, Donald S.
Burdick and Kong Chu. 1968. Computer Simulation
Techniques. John Wiley and Sons, Inc., New York,
322 pp.

APPENDICES

APPENDIX A

Based on the studies of Lewin (11) the development

of the new fruits was hypothesized to take the following

form:
1 Nc-1(T)
+ = + -_ . O O .
XC(T ) (a b WC_1(T) + c TX + d TN + e DT) Nc_1(T)
2
TX = (TMAX - 85)
TN = (TMIN - 65)2
DT = (TMAX * TMIN - 20)2
Where:

X = number of new fruit of size grade c
Nc = number of old" fruit of size grade c

W = weight of old fruit of size grade c

TMAX = maximum temperature between the sampling times,
T and T+1
TMIN = minimum temperature between the sampling times,

T and T+l.

The constants 85 and 65 in TX and TN are respectively
the most favorable phototemperature, and nyctotemperature
for cucumber fruit enlargement. The constant 20 in DT is
the estimate of diurnal thermoperiodicity which produced

maximum growth rate as found by Lewin (11).

 

*"01d" as defined in 3.5.10.

106

107

The parameters a, b, ... , e in the above equation
were estimated by multiple regression utilizing the method
of stepwise deletion of variables. The final results of
such a run were presented in Section 3.7.

The independent variables TX, TN, DT and NC_1/WC_1
were deleted from the final model for the following reasons:

(1) The sign of their coefficients were inconsistent
with results of previous investigations.
(2) Each of them had a significance probability level

above 0.01.

(3) The variability in fruit development explained by
them was very small.

Changing parameters in TX, TN or DT made little

difference in result as compared to the one above.

APPENDIX B

The matrices A, B and §_in 3.9.1 and 3.9.2 for a
simulation model based on size grades 1, 2, 3 and 4 were

estimated to be as follows:

 

 

 

 

_.
1 0.13 0 o
o 0.87 0.30 0
A =
o o 0.70 0.20
0 0 0 0.80
F- ‘7
0
U(T) = 6
0
M. s M(T) s M
a = 0 1 f
M. = 0.50
l J.
_ .5 Mf = 0.25
_ 0.485 0 o 0 7
0 0.205 0 0
E = 0 0 0.127 0
0 0 0 0.035
B- __

 

 

108

APPENDIX C

If the parameters in A_in 3.9.1 are estimated

directly from the observed sample values they will be

biased and inconsistent. The estimated parameters

result from ordinary least squares are as follows:

1.02 0.23 0.08 0 0
0 0.44 0.49 0 0
0 0 0.70 0.24 0
A:
0 0 0 0.64 0.36
0 0 0 0 0.82
0 0 0 0 0

 

Notice that the columns do not sum to one.

that

CO

C

This

implies that the model structure in 3.5.1 is not satisfied.

109

 

 

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