MECHANICAL HARVEST SYSTEM. smumon AND _‘ —’
DESIGN CRT’IERIA FOR PRO€£S$£NG WES _ ‘ .

Thesis for the Degree of Ph. D. ‘ 7
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GALEN KENT BROWN
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This is to certify that the
thesis entitled
Mechanical Harvest System Simulation and
Design Criteria for Processing Apples
presented by I

Galen Kent Brown

has been accepted towards fulfillment
of the requirements for

Ph-D- _degree in W1
Engineering

 

 

 

#W

Major professor

Date w

0-7639

ABSTRACT

MECHANICAL HARVEST SYSTEM SIMULATION
AND DESIGN CRITERIA FOR
PROCESSING APPLES

BY

Galen Kent Brown

The primary objectives of this study were to design
a harvest simulation model for use in evaluating mechanical
harvest systems proposed for processing apples and to
deve10p a technique for specifying the needed values of the
harvest system design parameters so that the harvest system
will be of economic benefit to nearly all of the growers
included in the simulation.

The four apple varieties with the largest processed
volume (standard type trees in acreage proportions deter-
mined from published data) were assumed to be representative
of production for the individual grower.

The discrete time form and a simulation time incre-
ment of one day are used in the harvest simulation model.
Known functional relations between yields, harvest cost,
and crop income were used to insure realism in the economic
behavior of the model. Separate stochastic models were

designed to describe annual yield; daily windloss; and

Galen Kent Brown

daily work time lost due to Sundays, rain, or machine
breakdowns.

The yield model for each variety was formulated as
a first-order autoregression relation having negative cor-
relation between yields for successive years. Using this
approach the biennial bearing tendency as well as the-
probability distribution of annual yield is adequately
described for harvest simulation. A mean value for daily
growth rate, estimated from growth data, was included in
the simulation model.

The daily windloss model for each apple variety was
formulated as a non-linear function of daily average wind
velocity. Parameters for this relation were derived using
iteration to satisfy the requirement that simulated wind-
loss must closely match the probability density function
and expected value for annual windloss. The parameter values
were different for each variety. Relations between windloss
and wind velocity were not available in the literature.

The model for daily lost work time consists of
separate models for including the effect of Sundays, rain,
or machine breakdowns. The first Sunday of each season
occurs at random within the first seven days, followed by
successive Sundays at seven-day intervals. The lost time
value for rain is generated using the cummulative distribu-
tion function for lost time, derived from historical records

of hourly rain observations and a no-work criteria based on

Galen Kent Brown

the amount of rain. The model for machine breakdowns
assumes that operating time between breakdowns is expo-
nentially distributed.

Harvestrate--acreage relationships were determined
for the Grand Rapids area of Michigan. In 90% of the
seasons, a harvest system with a harvest rate of 10 trees
per hour will be able to handle up to 70 acres of standard
type apple trees. A change in acreage requires a prOpor-
tionate change in harvest rate.

A general policy of delaying the start of harvest
in order to increase the harvested volume, anticipating
that fruit sizing may be greater than windloss, was evaluated
by simulation. This policy was not beneficial because the
expected value of harvested volume decreased, except under
conditions of sustained rapid fruit growth.

The harvest simulation model HARVSIM was designed to
evaluate proposed mechanical harvest systems. A simulation
for the period 1968-1971 showed that expected margin (a
measure of economic benefit) for two assumed mechanical
harvest systems increased while the probability of negative
margin decreased each year.

A sensitivity analysis was performed to identify
critical parameters in the design or Operation of the
harvest systems. The analysis showed that small variations
in harvest rate, machine recovery, and fruit injury, for

these mechanical harvest systems cause large variations

Galen Kent Brown

in expected margin and expected planning margin. The
analysis also showed that similar results occur for
variations in yield and hand picking cost--thus these
uncontrollable parameters must be closely estimated.

Three criteria were examined for use in selecting
design parameter values for mechanical harvest systems.
A low probability of negative margin (in the range of 10—20%)
is proposed as a design criteria because it will provide
the highest level of assurance that a proposed harvest
system will be of economic benefit.

For convenient use in harvest system design, the
relations between the probability of negative margin and
various combinations of design parameter values can be

described graphically. This procedure is illustrated.

     
 
 
 

Approved:
ajor Professor

Approved:

 

epartment Chairman

MECHANICAL HARVEST SYSTEM SIMULATION
AND DESIGN CRITERIA FOR

PROCESSING APPLES

BY

Galen Kent Brown

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Engineering

1972

j ACKNOWLEDGMENTS

The author wishes to express his sincere appreciation
to the following persons and organizations who contributed to
this study:

To Dr. J. B. Holtman, my major professor, for counsel
and guidance provided during the entire graduate program.

To the other members of my guidance committee, Dr.

D. H. Dewey (Horticulture Department), Dr. D. J. Ricks
(Agricultural Economics Department), and Dr. C. J. Mackson
(Agricultural Engineering Department), for their helpful
suggestions and continued interest.

To District Extension Horticultural Agents Frank
Klackle (Grand Rapids) and Stewart Carpenter (Paw Paw); to
apple growers David Friday (Hartford), Milo Brown (Martin),
and Bill Brahman (Belding); and to processor representatives
Ray Floate (Michigan Fruit Canners) and Jim Mortenson
(National Fruit Product Company) for the information they
contributed based on their experience.

To the Agricultural Engineering Research Division,
Agricultural Research SErvice, U.S. Department of Agriculture
for financial assistance (Government Employees Training Act)
and confidence; and to Jordan Levin, Investigations Leader,
Fruit and Vegetable Harvesting Investigations, for his
counsel, encouragement, and assistance.

ii

To my wife Ann and daughters Joscelyn and Teri
for their love, understanding, encouragement, and

sacrifice.

iii

 

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . .

LIST OF TABLES . . . . . . . . . . .

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

Chapter

1.

NNNNNNNNN

INTRODUCTION . . . . . . . . . .

1.1 General Information . . . . . .
1.2 Need for Harvest Mechanization . .
1.3 Need for a Simulation Model . . .
1.4 Objectives of the Study . . . .

FORMULATION OF THE SIMULATION MODEL . .

.l Controllable Inputs . . . . . .
.2 Uncontrollable Inputs . . . . .
.3 Environmental Inputs . . . . .
.4 Model Outputs . . . . . . . .
.5 Design Parameters . . . . . .
.6 Criteria for Measuring Goodness . .
.7 Mathematical Relations in the Model
.8 Assumed Grower Characteristics . .
.9 Stochastic Simulators Required . .

YIELD SIMULATOR . . . . . . . . .

3.1 Available Data . . . . . . .
3.2 Model Formulation . . . . . .
3.3 Parameter Estimation . . . . .
3.4 Yield Model Characteristics . . .
3.5 Daily Fruit Growth . . . . . .

3.5.1 Available Data . . . . .

WINDFALL SIMULATOR . . . . . . . .

4.1 Available Data . . . . . . .
4.2 Model Formulation . . . . . .
4.3 Parameter Estimation . . . . .
4.4 Model Verification . . . . . .

iv

Page
ii
vi

vii

I—J

\lmU'lH

Chapter
5. HARVEST RATE--ACREAGE RELATIONSHIP

5.1 Available Data . . . . .

5.1.1 Length of the Harvest Season

2
3 Harvester Capacity . .
4

Accumulated Working Hours

5.1.5 Work Time Lost Due to Rain

5.2 Maximum Acreage Policy . . .
5.3 Delayed Harvest Policy . . .
5.4 Harvest Policy Selection . .

6. HARVEST SIMULATION MODEL . . . .

.1 Initialization . . . . .
.2 Subroutine YIELD . . . . .
.3 Subroutine WIND . . . . .
.4 Random Number Array .
.5 Subroutine GROWTH .

.6 Subroutine WINDLOSS . . . .
7 Subroutine WORK . . .

8 Subroutine HARVEST . .

9 Subroutine GROSUM . . . .
10 Subroutine FINAL . . .
.11 Output . . . . . . . .

7. HARVEST SIMULATION . . . . . .
7.1 Simulation Assumptions . . .
7. 2 Simulation Results . . . .
7. 3 Sensitivity Analysis . .
7. 4 Harvest Simulation Conclusions

8. APPLICATIONS OF SIMULATION RESULTS

Varieties and Respective Acreage

8.1 Criteria for Selecting Design Parameter

Values . . . . . . .

2

3
Values . . . . .

4 Additional Relations . . .

5

9. SUMMARY AND CONCLUSIONS . . . .

10. RECOMMENDATIONS . . . . . . .

REFERENCES 0 O O O O O O O O O O

Probability--Parameter Relations

0

Advantages, Disadvantages, Limitations

Procedure for Selecting Design Parameter

Page
49

50
50
50
51
51
52
53
55
6O

62
62

65
65
65
66
66
66
67
67
67

68
68
75
80
89
97
98
99
106
109
110
112
117

119

Table

LIST OF TABLES

Michigan Apple Production and Utilization

Determination of Acreage Proportions
Estimated Annual Yields . . .

Annual Yield Data . . . . . .
Biennial Bearing Statistics . . .
Statistics for Annual Yield Data .
Statistics for Assumed Yield pdf's .
Growth Rate Statistics . . . . .
Estimated Annual Windloss pdf's . .
Length of Harvest Period . . . .
Wind Velocity Model Parameters . .

Daily Windloss Model Parameters . .

Harvest Starting Dates, Grand Rapids Area

Harvest Rate--Acreage, Maximum Acreage Policy

Idle Days Between Varieties , Maximum Acreage Policy

Daily Work Time Lost Due to Rain .
Delayed Harvest Results, McIntosh .

Delayed Harvest Results, Jonathan .

Assumed Crop, Price, and Labor Cost Parameters

Assumed Harvest System Parameters .

Simulated Margin and Harvest Cost .

Simulated Productivity and Volume Ratios

Risk of Negative Margin . . . .

vi

Page

18
21
22
24
28
28
33
37
39
41
47
51
54
54
56
58'
59
72
73
76
77

80

LIST OF FIGURES

Figure Page
3.1 Annual Yield pdf's . . . . . . . . . . 31

4.1 Daily Average Wind Velocity pdf's and CDF's . . 38

4.2 Response Surface for McIntosh Windloss . . . 43
4.3 Response Surface for Jonathan Windloss . . . 44
4.4 Response Surface for Golden Delicious Windloss . 45

4.5 Response Surface for Northern Spy Windloss . . 46
6.1 Simplified Flow Chart of HARVSIM . . . . . 63
7.1 Sensitivity to Equipment Cost . . . . . . 82
7.2 Sensitivity to Crew Size . . . . . . . . 82
7.3 Sensitivity to Harvest Rate . . . . . . . 83
7.4 Sensitivity to Machine Recovery . . . . . . 83
7.5 Sensitivity to Fruit Injury . . . . . . . 83
7.6 Sensitivity to Total Acreage . . . . . . . 84
7.7 Sensitivity to Yield . . . . . . . . . 84
7.8 Sensitivity to Fruit Size . . . . . . . .‘ 85
7.9 Sensitivity to Natural Defects . . . . . . 85
7.10 Sensitivity to Fruit Price . . . . . . . 85
7.11 Sensitivity to Labor Cost . . . . . . . . 86
7.12 Sensitivity to Picking Cost . . . . . . . 86
7.13 Sensitivity to Equipment Cost . . . . . . 90

7.14 Sensitivity to Crew Size . . . . . . . . 90

vii

Figure Page
7.15 Sensitivity to Harvest Rate . . . . . . . 91
7.16 Sensitivity to Machine Recovery . . . . . . 91
7.17 Sensitivity to Fruit Injury . . . . . . . 91
7.18 Sensitivity to Total Acreage . . . . . . . 92
7.19 Sensitivity to Yield . . . . . . . . . 92
7-20 Sensitivity to Fruit Size . . . . . . . . 93
7-21 Sensitivity to Natural Defects . . . . . . 93
7.22 Sensitivity to Fruit Price . . . . . . . 93
7.23 Sensitivity to Labor Cost . . . . . . . . 94
7.24 Sensitivity to Picking Cost . . . . . . . 94

8.1 Design Parameter Values vs. Probability of
Negative Planning Margin . . . . . . . 101

8.2 Design Parameter Values vs. Probability of
Negative Margin . . . . . . . . . . 102

8.3 Design Parameter Values vs. Probability of
Negative Margin . . . . . . . . . . 104

8.4 Design Parameter Values vs. Probability of
Negative Margin . . . . . . . . . . 105

viii

1 . INTRODUCTION

Apple growers in Michigan are interested in feasible
mechanical harvest systems for processing apples. To
determine if a mechanical harvest system will be feasible
requires knowledge of its economy of Operation under the
many harvest conditions a grower may encounter. This informa-
tion could be obtained, after considerable expense and time,
by Operating actual harvest systems under many conditions.
However, a simulation model could provide similar informa-
tion at less cost and in less time.

This study is intended to define the needed informa-
tion and develop specific models or procedures so that
simulation can be used to evaluate and design mechanical
harvest systems for processing apples. The procedures used
are expected to be applicable to several other fruit and

vegetable crops.

1.1 General Information

 

A 1968 survey of commercial orchards in Michigan
showed nearly 3,450,000 apple trees of which 69% were the
standard type and 31% were the semi-dwarf or dwarf type (18).
This survey included 55,000 acres of bearing trees with a

production of 535,000,000 pounds having a value of

$28,083,000. Over 80% of the Michigan apple industry was
located in the western half of the lower penninsula and
within about 40 miles of the Lake Michigan shore line.

In 1970 the Michigan Apple Commission had a mailing
list of 2000 growers and estimated that 500 of these growers
produce 80% of the State's apples.l These figures suggest
the "average" grower had about 27.5 acres of bearing apple
trees while 80% of the production came from farms averaging
88.0 acres of bearing apple trees. Kelsey (14) states:
"Commercial fruit production is being concentrated into a
relatively small number of farms with larger acreages--
standards for a satisfactory income on a tree fruit farm
might be 100 acres of bearing fruit--."

During the period 1965-1969, 53-58% of the total
apple production in Michigan went into processed utiliza-
tion (19). Michigan Apple Commission data, Table 1.1,
show that more than 50% of the annual production of 6-8
apple varieties went into processed utilization in 1969
and 1970. Formerly, apples were grown and harvested for
fresh utilization with the lower quality fruit being
graded out for processed utilization. Recently, an
increasing number of growers are growing some apple

varieties strictly for processed utilization.

 

1Personal communication, 0. D. Pynnonen, Michigan
Apple Commission, East Lansing, Michigan, August 31, 1970.

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The yearly prices offered by processors for 20
different apple varieties, as published by the Michigan
Agricultural Cooperative Marketing Association (4), are
not constant from year to year. The price Offered depends
on many factors, among which are apple production in
primary processing states, carryover stocks of processed
products, United States disposable income, and processor
margin (24, 27).

Apple size as well as quality are important in
determining the price received by a grower. Apples must
usually be ZH-inch diameter or larger to bring the top
price for processing. Apples smaller than this are
presently used for juice or cider and bring a price 50-60%
less than for the larger size. The grower is not paid
for defective apples which result from excessive bruises,
punctures, or certain natural defects.

The size of crOp varies from year to year depending
on weather conditions during the pollination period, size
of crop the preceeding year, and other factors.

Weather conditions during the harvest season can
have a major affect on the quantity and quality of the
crop, and can be expected to be different for each major
apple growing area in the state. Windstorms during
harvest may cause a large portion of the crop to become
windfalls, which usually have very low value. Water

supply (rain) during the harvest season can affect the

growth rate of apples, and thus the total yield and size
distribution of the crop. Rain also results in lost work
time and thus affects the harvest rate--acreage relation

for any harvest method.

1.2 Need for Harvest Mechanization

 

The number of workers on Michigan farms decreased
at an average annual rate of 5.5% during 1955—65 (28).

Many fruit growers have difficulty Obtaining enough pickers
to harvest their fruit at the proper time.

Hired labor is recognized as a major item of cost
in fruit production and is often greater than 50% of the
total cost of production (29). There is, however, a wide
difference between a worker's average hourly earnings in
agriculture and in manufacturing. If harvesting was
mechanized, increased productivity might permit hourly
wages to be increased and would reduce the peak demand
for harvest labor.

The trend toward growing more apples strictly for
processed utilization, the need to increase worker pro?
ductivity, and a need to increase grower income all
reinforce the interest in mechanical harvest methods for
apples. Several state and federal research agencies in
the United States are now working on the development of
mechanical harvest methods for apples, as well as apricots,
avocados, cherries, peaches, pears, oranges, grapefruit,

lemons, and papaya.

1.3 Need for a Simulation Model

 

Analytical techniques for estimating the necessary
or allowable values for mechanical harvester performance
parameters have been proposed and used for prunes,
apricots and cling peaches (9), citrus (5, 13, 33), apples
(30), and various fruits and vegetables (10). All of
these techniques have used average values for the variables,
with the performance parameters computed for a breakeven
condition (based on harvest costs or grower returns)
between the conventional hand harvest method and a proposed
mechanical harvest method.

A stochastic simulation model for evaluating the
effect of harvest policy, labor cost, fruit sale price, and
fruit ripening rate on expected net return to papaya
growers was developed by Wang, et_§l, (32). Such a model
has not been developed for apple growers, although many of
the variables in apple production, harvesting, and marketing
are of a stochastic nature. A simulation model is needed
for evaluating conceivable mechanical harvest systems and
specifying the necessary values for many of the design
parameters. Such a model will provide valuable information
not presently available to researchers, designers, growers,

and processors.

1.4 Objectives of the Study

 

As an initial step toward meeting the above need,

the following objectives were selected.

1.

Design a harvest simulation model which can be used

to evaluate the economic benefit of mechanical harvest
systems for an apple grower whose production is strictly
for process utilization.

Design the necessary sub-parts of the above simulation
model which are required to describe yields, important
weather occurrances during harvest (wind, lost time due
to rain), and harvest rate--acreage relationships.
Define outputs from the simulation model which provide
planning information not presently available to
researchers, designers, growers, and processors.
Evaluate a hypothetical mechanical harvest system, for
illustration purposes, using available information and
the simulation model.

Develop a technique for specifying the needed values of
the harvest system design parameters so that the risk
of negative margin for a mechanical harvest method
compared to the hand harvest method is at a particular
probability level. Margin = [(income from mechanically
harvested crop-rmechanical harvest cost) - (income from

hand harvested crop-hand harvest cost)].

2. FORMULATION OF THE SIMULATION MODEL

The objectives of this study are stated in general
terms at the end of Chapter 1. Formulation of the simula-
tion model to meet these objectives involved consideration
of both the type of outputs desired and the type of inputs
available.

Simulation 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
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 (17). The outputs (defined in
Section 2.4) are intended to provide both annual and plan-
ning period (i.e., the period of years required to payoff
the machine investment) information. Some input informa-
tion is available on a daily basis and some is available
on an annual basis. After considering all of these facts,
the discrete time form was selected with a simulation time

increment of one day.

Accurate output information requires both accurate
input data and the use of accurate functional relations in
the model. Annual input data on fruit quality, fruit value,
labor costs, and equipment costs are available from
historical records, as are daily input data on weather
conditions. Known functional relations between yields,
harvest cost, and crop income were used to insure realism
in the economic behavior of the model.

To include all possible interactions and inputs in
the model formulation, a very complex mathematical model
would be required. In general, the mathematical model
should be formulated to yield reasonably accurate descrip-
tions or predictions of the behavior of a given system
while minimizing computational and programming time (22).
Thus, boundaries were imposed on the scope of the defined
problem. Factors such as: costs not incurred in the
harvest operation; management ability of the grower;
losses due to labor shortages, strikes, or carelessness;
and alternate investment strategies were not included in
the model formulation. These types of cost factors are
difficult to quantify and, in the opinion of the author,
are not required for the relative evaluation of mechanical
harvest systems.

The many variables, parameters, outputs, and other

factors of importance in the mathematical model were grouped

10

as suggested by Asimow (1). These groups are discussed in

the following

These
the equipment
ECOST
ELIFE
OPER
HRATE
RH
RP

HI

These
the equipment
NVAR
ACRES
TREED

TII

TRIKE

six Sections.

2.1 Controllable Inputs

 

inputs were assumed to be controllable by
designer, the equipment operator, or both.

- Initial equipment costs, dollars

- Equipment life, years

- Crew size, number of workers

- Harvest rate, trees per hour

- Machine recovery, portion of on-tree yield
- Pickup recovery, portion of on-ground yield

- Fruit injury, portion of harvested volume

2.2 Uncontrollable Inputs

 

inputs were assumed to be uncontrollable by

designer, operator, or both.

- Number of varieties considered

- Acres of producing trees for each variety

- Tree density, trees per acre

- Taxes, insurance, interest, expressed as
a fixed proportion of initial equipment
cost

- Cost of repairs, fuel, taxes, dollars
per hour

- Tractor cost, dollars per hour

11

2.3 Environmental Inputs

 

These inputs were assumed to be solely the result

of climatic or
by the grower.

Y
o

Y.
1

YW.
1

WL.
1

GR

PG

DNAT

ROT

PRICE

OCOST

PCOST

economic conditions not highly controllable

Initial on-tree yield, bushels per tree
On-tree yield, day i, bushels per tree
On-ground (windfall) yield, day i,
bushels per tree

Windloss for day 1, portion of on-tree
yield

Growth rate, daily increase in on-tree
yield

Portion of the crop 28-inch diameter

and larger

Portion of the crop with natural defects
Portion of windfall fruit which is
decayed

Dollars per hundredweight (cwt) for fruit
of processing, juice, or drop quality
Labor cost, dollars per man-hour for
mechanical harvest

Handpicking cost, dollars per bushel

2.4 Model Outputs

 

The following outputs were considered to be suffi-

cient to provide the necessary information required to

meet the objectives of the study.

12

YMAR - Margin, dollar difference per year between
mechanical harvesting and hand harvesting

EMAR - Expected value of YMAR

SDMAR - Standard deviation of YMAR

PMAR - Planning margin, sum of YMAR during the
planning period

EPMAR — Expected value of PMAR

SDPMAR - Standard deviation of PMAR

HC - Harvest cost, dollars per bushel

EHC - Expected value of HC

SDHC - Standard deviation of HC

PROD . - Productivity, bushels mechanically
harvested per man-hour

EPROD - Expected value of PROD

SDPROD - Standard deviation of PROD

PR - Ratio of processing volume for mechanical
harvest compared to hand harvest

EPR - Expected value for PR

SDPR - Standard deviation of PR

JR, EJR, SDJR - Ratio, expected value, and standard
deviation for juice volume

DR, EDR, SDDR - Ratio, expected value, and standard
deviation for windfall volume

The probability of negative margin can be calculated

from EYMAR, SDYMAR, and the probability density function

13

(pdf) for YMAR. A similar procedure can be used for cal-

culating the probability of negative planning margin.

2.5 Design Parameters

 

The researcher, faced with the problem of evaluating
various methods for mechanically harvesting apples, must
determine acceptable values for all of the Controllable
Inputs, listed in Section 2.1, considering the constraints
placed on all parameters in the system. For this reason
all of the Controllable Inputs in this study are considered
to be design parameters.

Other factors, not explicitly stated here, can also
be considered as design parameters due to their direct
influence on some of the Controllable Inputs. Examples of
such factors would be: tree modification to decrease fruit
injury, increase fruit removal, and increase harvest rate;
the use of special chemicals to reduce wind losses, increase
fruit removal, or decrease fruit decay; and shared use of
harvest equipment with other crops such as cherries,
peaches, pears, and plums so that the relative equipment
cost can be reduced. However, because such factors are
still experimental or require particular conditions they

will not be analyzed in this study.

2.6 Criteria for Measuring Goodness

 

Establishing exact goodness criteria is a difficult

task which will certainly result in different criteria if

14

left to the various interest groups involved. However,

general guidelines suggest that the following conditions

should exist:

1. The probability of negative margin and planning margin
should be low.

2. Worker productivity should be high enough so that labor
requirements for harvest are compatible with year-
around requirements and worker hourly earnings are
comparable to those in manufacturing.

3. The expected volumes of fruit in various quality cate-
gories should be within manageable limits for the

grower and processor.

2.7 Mathematical Relations in the Model

 

The volume of fruit available for harvesting from
each tree is determined for the first day of harvest and
is then adjusted daily to reflect the effects of windfalls
and fruit growth, according to the recursive relations:

Y1 = Y0 (1 - WLl)

K
II

2 Y1 (GR)(1 - WL2)

Yi + 1 = Yi (GR)(1 - WLi + l)
The accumulative volume of fruit available for

harvesting from the ground at each tree as windfalls is

determined by the relations:

15

YWl = YO (WLl)
YW2 = Y1 (GR)(WL2) + YWl
YWi + l = Yi (GR)(WLi + l) + YWi

The volume of fruit harvested from the tree or
picked up from the ground is determined with the following
two equations:

Volume from tree = Yi (RH)

Volume from ground = YWi(l-ROT) RP

The volume equations are multiplied by the number
of trees harvested each day, then summed daily to determine
total volume for each season. The number of trees harvested
each day is determined with the following equation:

Trees harvested = HRATE X HT
Harvest time, HT, is determined daily and depends upon the
occurrence of a Sunday, lost time due to rain, or a.
mechanical breakdown.

Hand harvest cost for picking fruit, or picking up
fruit, is calculated by converting the volumes to hundred-
weight (cwt), then multiplying by the appropriate picking
cost.

Mechanical harvest cost is calculated using the
following relation:

0.90 TII

Annual mechan1cal harvest cost = ECOST (EL FE + —§—)

 

+(RFO + TRCOST + OPER X OCOST) HTIME

16

This relation assumes straight-line depreciation, a salvage
value of 10% of the initial equipment cost, and that main-
tenance and Operating costs are directly proportional to
the annual hours of operation (HTIME). HTIME is accumulated
for each harvest method as the simulation proceeds through
each season.

Three categories Of fruit are assumed to have
economic value. The volume of fruit per tree in each
category is determined with the following three equations:

Processing = Yi(PG-DNAT)(l-HI) RH

Juice Yi[(l-PG) - DNAT] (l-HI) RH

Windfall

YWi (l-ROT) RP

The grower is paid processing prices for fruit
which is 2k-inch diameter or larger and is not a windfall,
unless it has natural defects or excessive harvest injury.

The grower is paid juice prices for fruit which is
less than 2k-inch diameter and is not a windfall, unless
it has natural defects or excessive harvest injury.

All fruit picked up from the ground are classed as
windfalls regardless of size or natural defects. If a
market exists for this category of fruit, the grower is
paid windfall prices for fruit which does not have decay
or excessive harvest injury.

Income is calculated by converting the total
volume in each category to cwt, then multiplying by fruit

value and summing over each category.

17

After completion of each season, the difference
between income and cost is computed for each harvest
method, then the difference for hand harvesting is sub-
tracted from the difference for mechanical harvesting to
determine margin. The expected value and standard devia-
tion of margin is computed for each year as is the
expected value and standard deviation of planning margin.
These statistics can be used to calculate the probability

of negative margin and negative planning margin.

2.8 Assumed Grower Characteristics

 

Michigan apple processors buy at least 20 apple
varieties and a typical Michigan grower may raise several
of these. For purposes of this simulation, the four
varieties with the largest processed volume (McIntosh,
Jonathan, Golden Delicious, Northern Spy) were assumed to
be a representative cross-section of the varieties used
for processing. In addition, a typical proportion of
total acreage was estimated for each variety using tree
population data published in the 1968 survey (18), and
average tree planting densities for standard type trees.

This information is summarized in Table 2.1.

2.9 Stochastic Simulators Required
Many of the Environmental Inputs are of a sto-
chastic nature. However, some are more important than

others in terms of their expected variations and affect

18

Table 2.1 Determination of Acreage Proportions.

 

 

Variety Numberl %§%E§2 Acres3 Acies
McIntosh 279,053 27 10,340 25
Jonathan 650,350 34 19,130 45
Golden Delicious 177,400 34 5,220 13
Northern Spy 193,308 27 7,160 _11
TOTALS 41,900 100

 

1Number of trees in Michigan. Taken from Table 7,
Michigan 1968 Fruit Tree Survey.

2Estimate of typical planting density for standard
type trees, obtained from Frank Klackle, District Extension
Horticultural Agent, Grand Rapids.

3Calculated from number of trees and typical
planting density.

on the outputs. For this simulation, initial annual yield,
daily windloss, daily work time lost due to rain or the
occurrence of Sundays, and the occurrence of a breakdown
of the mechanical harvester were modeled as stochastic
processes. The design of each model is discussed in

detail in a following chapter.

3. YIELD SIMULATOR

Annual yields for all apple varieties vary from
year to year. Hoblyn, gt_al. (12) studied the fruiting
habits of 15 varieties of apple in England. They proposed
two constants, B and I, which could be calculated from
yield records to describe the biennial bearing tendency.
The constant B (on a scale from zero to 100) indicates
whether the variety is wholly, partially, or not at all
biennial. The constant I (on a scale from zero to 1.0)
indicates the magnitude of the yield fluctuations. Their
study showed B values from 61 to 91 and I values from
0.26 to 0.71, with grand means of 74.3 and 0.48 respectively,
for 12 years of data on the 15 varieties. Wilcox (34)
conducted a study of correlations between tree growth and
fruiting and found that fruit set (thus yield) one year had
a highly significant negative correlation with set (thus
yield) the following year. Singh (25) reported that alter-
nate bearing of certain fruit plants (including apple)
is a major problem in commercial fruit growing all over
the world. He listed 125 references relating to some

aspect of the alternate bearing problem.

19

20

Annual apple yields are recognized to be stochastic
in nature, alternating from high to low depending on the

variety and geographic location of the planting.

3.1 Available Data

 

Accurate records of total annual yield (volume
picked + volume windfall) by variety are not maintained
by most growers. Instead, the grower has a mental record,
or impression, of what his minimum, most likely (typical),
and maximum annual yield has been for each variety. Based
on replys from five growers and horticultural agents,
values for the above classes of yield were determined,
Table 3.1.

Yield records for a typical planting of the four
apple varieties were obtained from the Graham Experimental
Station at Grand Rapids, Table 3.2. Using these data to
determine means and standard deviations would not be
statistically desirable because the data covers a short
time period and this planting was only one of many similar
plantings in the Grand Rapids area. However, these data
were used to: (1) calculate biennial bearing tendency;

B, and intensity, I: (2) estimate the correlation between
successive annual yields; and (3) estimate the relative
size of the standard deviation of annual yield for each

variety.

21

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+ OOMOOQ madao>v OOHOOa Hmsccm mo coacflmo ucmmm HmuouHSOODMon cam umzoum so Oommm

 

 

 

 

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mom humaum>
ouom Mom mamnwsn .Uamwm Hmsccfi momma menu Mom mamnmsna.©HOHM Hmsccd

 

.mOHwOs amazes Omumsflumm .H.m manna

22

Table 3.2. Annual Yield Data.

 

Annual Yield,l pounds per tree

 

 

Year McIntosh Jonathan Golden Delicious Northern Spy
1948 0 0
1949 15 0
1950 50 40 4O 0
1951 87 72 91 2
1952 142 132 65 2
1953 184 86 119 38
1954 259 133 76 58
1955 121 91 347 296
1956 514 277 146 78
1957 176 248 556 291
1958 650 305 125 292
1959 407 565 630 504
1960 1126 508 217 284
1961 1170 709 866 1112
1962 699 602 349 19
1963 926 538 965 695
1964 1158 766 381 504
1965 443 439 979 880
1966 1086 625 168 126
1967 398 430 947 658
1968 823 879 531 358
1969 1112 566 943 1044
1970 830 994 386 468

 

lAverage for Hibernal and Seedling interstocks,

Block 2, Graham Experimental Station, Grand Rapids,

Michigan.

23

3.2 Model Formulation

 

For harvest simulation, a time series Of either
historical yields or stochastically generated yields is
required. Historical records have the disadvantages of
limited length and number. For these reasons a stochastic
model was developed for generating an annual yield time
series.

The Graham Station yield data were plotted and the
first year of commercial production was estimated.
Commercial production is characterized by fairly level
production over a period of years. The biennial bearing
tendency and intensity, Table 3.3, were calculated using
yield data for the first and succeeding years of commercial
production. The results show that during the 9-12 years of
commercial production, the biennial bearing tendency was
very strong in McIntosh and Jonathan and complete in
Golden Delicious and Northern Spy. The intensity was
between 0.27 and 0.55 for all varieties.

Yield each year is an integrated result of many
factors some of which may be tree age, tree surface area,
variety, nutrition, water supply, frost at pollination
time, amount of hand or chemical thinning, and yield the
previous year. For the purpose of constructing a usable
yield simulator, it was hypothesized that successive yields
during commercial production (no growth trend present)

could be described by the first-order lag model:

24

Table 3.3. ‘Biennial Bearing Statistics.

 

 

Variety Years1 ‘82 I3 64 p5
McIntosh 1959-70 70 0.27 -0.45 -0.45
Jonathan 1961-70 88 0.34 -0.60 -0.55
Golden Delicious 1962-70 100 0.47 -0.93 -0.80
Northern Spy 1961-70 100 0.55 -0.77 -0.70

 

lYears representative of commercial production,
Table 3.2.

2Biennial bearing tendency =

Number of pairs of years with sign of (Yi+ - Yi) different

1
Total number of pairs of years

 

IY1+1 ‘ Yil
1+1 + Yi

3Biennial bearing intensity = Average of

 

4Estimate of the correlation between annual yields

Y?i and Yi-l’ after adjusting for any growth trend present in
t e data.

5Assumed correlation between Yi and Yi-l' used in
the autoregressive yield model.

Yi=p+p(Yi_l-p)+ei
Where:
Yi = Annual yield, year 1
1-1 = Annual yield, year i-l
u = Mean annual yield
9 = Correlation between Yi and Yi—l
e. = Random disturbance term, year i

25

A slight growth trend is suggested in the commercial
production yield data for all varieties, Table 3.2, so
perhaps production had not yet leveled off for these trees.
Prior to estimation of p, the linear time trend was

removed by regressing Yi on time using the model:

Y. = a + Bt. + v.

1 1 1

Where

Yi = Annual yield, year 1

ti = Year number corresponding to Yi
a = Y. intercept at t

1 o

B = Slope relation between Yi and ti

vi = Random disturbance term, year i

Residuals from this model were calculated for each

Yi’ then D was estimated by the product-moment method (26):

 

 

n
X r. r.
. i=2 1 1’1
D
n n
X r: V/; ri_l
i=2 i=2

Where:

D)
II

Estimate of correlation between Yi and Yi-l

r. = Residual for Y
model

i’ from linear time trend

26

ri-l = Residual for Y._l, from linear time trend
model 1
n = Number of residuals available

The results, Table 3.3, show that all correlations
are negative. A statistical test of the null hypothesis

H : p = 0 against the alternative hypothesis H p < 0

o 1'
cannot be applied to these coefficients because their
distribution is not known. Furthermore, the basic assump-
tion in correlation analysis of independence between
successive dependent variables has been violated by the
hypothesized first-order lag model.

The correlations were calculated from one set of
data covering a relatively short time, and thus could be
inaccurate estimates of the true correlations. Grower
opinion suggests that the relative order of correlation
magnitudes among varieties should be as calculated.

To represent the correlation between successive
annual yields for a commercial size grower, and mature
trees, the p values shown in Table 3.3 were assumed. These
are somewhat subjective, but are thought to be reasonable.

To generate random yields having a specified mean,
standard deviation, and correlation between successive
values, as hypothesized in the first-order lag model,
an equivalent first-order autoregression model described

by Llewellyn (16) was used. The general yield model for

each variety is given by:

27

Yi = u + p (Yi_l - u) + o (l - 02)3 Xi
Where

Y1 = Correlated annual yield per tree, year i

Yi-l = Correlated annual yield per tree, year i-l

p = Correlation between Yi and Yi-l

u = Mean of Yi

o = Standard deviation of Yi

Xi = Standardized random variable calculated from:
X: (lg—E)

Where: Y = Random (uncorrelated) value for
annual yield generated from the
cummulative distribution function
(CDF) for annual yield

u, o = Same as above

The random disturbance term, ei, in the first-order

lag model corresponds to the 0(1 - p2)15

Xi term in the
autoregression model.
The parameters u and o and the CDF need to be

specified before the autoregression model can be used.

3.3 Parameter Estimation

 

Estimates of the parameters u and 0 were calculated
using assumed yield pdf's based on the estimated minimum,
most likely, and maximum annual yields, Table 3.1. The
final pdf's selected, and corresponding estimates for u and

o, are given in Figure 3.1 and Table 3.5. The mean yields

28

Table 3.4. Statistics for Annual Yield Data.

 

 

Time'Periodl Mean2 (Standard Deviation2
McIntosh 1959-1970 20.2 ' 7.14
Jonathan 1961—1970 15.6 4.36
Golden Delicious 1962-1970 15.0 7.78
Northern Spy 1961-1970 14.0 8.67

 

lYears representative of commercial production,
Table 3.2.

2Bushels per tree.

Table 3.5. Statistics for Assumed Yield pdf's.

 

 

 

 

Yield per Treel Yield per Acre

Mean Standard Mean Standard

DeV1at1on DeV1at1on
McIntosh 17.16 4.41 463 119
Jonathan 14.12 3.04 480 103
Golden Delicious 13.47 3.63 458 123
Northern Spy 14.19 5.55 383 150

 

1Bushels per tree assumed for the yield models.

29

for the assumed pdf's are somewhat less than for mature
trees at the Graham Station, Table 3.4, but are felt to

be more representative of a long-time average. Kelsey,
Harsh, and Belter (15) state that a yield of 400 bushels
per acre (all varieties) would be a representative average
for an above average apple grower. The mean yields for
the pdf's are 16-20% above that average, except for
Northern Spy, because processed utilization is assumed and
windfalls are initially included in annual yield generated
from the pdf's. Mean yield for the Northern Spy pdf is
less than 400 bushels per acre because this variety can
have nearly zero yield some years, and has below average
yield for many growers.

The standard deviations for the assumed pdf's are
less than for the Graham Station data. The Station data
represent a small sample and thus could be unrepresentative.
However, the relative magnitude of the pdf standard
deviations are in the same order as for the Station data
and are felt to be realistic based on grower Opinion of
yield variation.

The pdf's were assumed to be composed of straight
lines because: (1) adequate observations of annual yield
are not available which allow plotting of yield histograms
and selection of a statistically rigorous shape; (2)
assuming a quasi-beta shape (a cosine curve from the

minimum to most likely and from most likely to maximum

30

values) for the pdf's resulted in a small standard devia-
tion; and (3) the straight-line pdf's can be easily altered
to change mean and standard deviation values.

Each assumed pdf was integrated to Obtain the CDF
of yield. The random annual yields, Y, required in the
autoregression model are calculated via inverse transforms
of uniform (0, 1) random numbers generated using a multi-

plicative congruential technique (22).

3.4 Yield Model Characteristics

 

Llewellyn (16) has described the task of determining
the true mean, variance, correlation coefficients of the
stochastic process, and distribution of the underlying
independent sequence, such as assumed in the first-order
lag model, as impossible. Thus, when generating a series
of autoregressive events it is advisable to be aware of the
pdf of the sequence being formed.

To estimate differences between the yield distribu-
tions based on minimum, most likely, and maximum values
and those formed by the autoregression model, computer runs
were made in which 5000 yield observations were generated,
and histograms were developed. The "grower" distributions
and "model" distributions are shown in Figure 3.1. The
mean and standard deviation values are almost equal but
the distribution shapes become progressively different as

the amount of negative correlation is increased. This

PROBABILITY 0F OCCURRENCE, %

IO

 

31

McIntosh

 

 

 

 

 

 

30
J
30
Golden Delicious . . -
_ L 1 1 ' Q L l- l
0 25 30
.Northern Spy
° .1 1 1 1 ° ' ° -4 -1
0 5 l0 IS 20 25 30

ANNUAL YIELD, bushels per tree

--- Estimated "Grower” pdf
. . . Autoregression “Model“ pdf

Figure 3.1 Annual Yield pdf's.

32

result is logical because if the correlation was -1.0
successive yeilds would have to be equidistant above and
below the mean. Thus, a symmetric distribution would

result.

3.5 Daily Fruit Growth

 

Fruit continue to grow after reaching the earliest
stage of maturity at which a grower can begin harvest. To
evaluate the change in yield per tree due to fruit growth
a value for daily growth rate during the harvest period is

needed. Growth rate will vary with variety and water

supply.

3.5.1 Available Data

 

Estimated optimum harvest dates for long-term
storage of McIntosh, Jonathan, and Red Delicious apples are
now published each year for Michigan (6). Data on fruit
weight before and during harvest, including the random
effect of water supply, are gathered from selected orchards
at weekly intervals for use in that study. From these
data, weekly weight for one sample of 20 apples at each
of four orchards in the Grand Rapids area were obtained
for the McIntosh and Jonathan varieties for the years
1969-1970 and 1968-1970 respectively. A daily growth rate
was computed for each orchard and weekly interval using

the relation:

33

GR = VW87W1
Where:
GR = daily growth rate

2
ll

observed weight on day 8

E
II

observed weight on day 1

This relation was derived from the recursive relation:

W2 = (GR) Wl
_ _ 2

W3 - (GR) W2 — (GR) Wl
_ 7

W8 — (GR) W1

The mean and standard deviation of GR, calculated
for all observations on each variety, are given in Table

3.6.

Table 3.6. Growth Rate Statistics.

 

Variety Mean Standard Deviation

 

McIntosh 1.0081 0.0115

Jonathan 1.0044 0.0074

 

34

Similar data were not available for the Golden
Delicious and Northern Spy varieties, so the statistics
for McIntosh were assumed to apply to both varieties. This
assumption was made because the three varieties have
similar size apples. Growth rate will have a minor
influence on the results of the harvest simulation, but is
required in order to evaluate the effect of delayed harvest
policies on harvested volume, and windloss.

After the crop is judged mature the volume of
fruit per tree available for harvesting can be adjusted
daily using the relation given in Section 2.7. For this
study the change in on-tree yield is assumed to be
directly proportional to the change in weight. The mean
value for growth rate, Table 3.6, will be used instead of

a randomly generated value.

4 . WI NDFALL SIMULATOR

The percentage of yield classed as windfalls, here-
after referred to as windloss, varies from year to year.
Hormone (stop-drop) sprays are available which may be
applied to the trees prior to harvest to reduce the expected
amount of windloss (3). Such sprays, containing Alpha-
naphthalenacetic acid (NAA) or 2,4,5-trichlorophenoxypropionic
acid (2,4,5-TP), have been used for many years by most
Michigan apple growers. A new material, succinic acid-2,2-
dimethylhydrazide (SADH or ALARR), is becoming increasingly
pOpular for certain varieties because among other desired
effects, it is an effective stop-drop and delays maturity
by 7-10 days. However, it is not clear if apples sprayed
with this material will adequately loosen for mechanical
harvesting. In developing this simulator, the proper use
of a conventional stop-drop spray has been assumed.

Annual windloss is the integrated result of
variety, wind occurrences, and the number of days required
to complete the harvest. To adequately simulate the wind-
loss process a stochastic model is needed for daily wind
velocity, the length of harvest period must be defined,
and a relationship between daily wind velOCity and daily
windloss must be developed.

35

36

4.1 Available Data

 

Accurate records of annual windloss are not main-
tained by most growers and although a literature search
provided some data, no long time records were found from
which a pdf for windloss could be constructed. Also, no
data were found relating daily wind velocity and daily
windloss.

Grower and Horticultural Agent opinion were used to
estimate the mean, maximum, and minimum annual windloss.
They thought that windloss for the McIntosh variety should
be about twice that for the other three varieties. In
addition, some impression about the shape of the annual
windloss pdf was provided by estimates of the number of
times annual windloss within specified intervals has
occurred during the past 20 years. These estimates are
given in Table 4.1.

Daily records for wind occurrences in the Grand
Rapids area are available from the U.S. Weather Bureau.

It was hypothesized that both magnitude and duration of
wind are important in determining the daily windloss.
For this reason the daily "Average Speed" (8), hereafter
referred to as daily average velocity, was selected for
use in the daily windloss model.

After reviewing the work of Murneek (21), Batjer and
Marth (2), Thompson and Batjer (31), and Edgerton and

Hoffman (7), the minimum daily windloss for McIntosh was

37

Table 4.1. Estimated Annual Windloss pdf's.

 

McIntosh

 

Annual Windlossl 0-10% 10-20% 20-30% 30P40% Over 40%

Probability 0.45 0.40 0.10 0.05 0.0

 

Jonathan, Golden Delicious, Northern Spy

 

Annual Windlossz 0-5% 5-10% 10-15% 15-20% Over 20%

Probability 0.45 0.40 0.10 0.05 0.0

 

lExpected value is 10-12%.

2Expected value is 5-6%.

assumed to be 0.2% and for the other varieties 0.1%.
Daily windloss greater than this was assumed to be the
result of daily average wind velocity greater than some
unknown base.

The nominal length of harvest period for each
variety was provided by individuals with considerable

experience in the apple industry, Table 4.2.

4.2 Model Formulation

 

The pdf for daily average velocity was constructed
for the period September 7-October 30 of 1951 and 1953-1970
at the Kent County Airport in Grand Rapids, Figure 4.1.

The means and standard deviations were calculated for every

38

 

 

 

 

l00
I2 P T
h C) Observed pdf 7
(5 Simulated pdf
l0 >- 0 Observed CDF — 80
__ A Simulated CDF
q
8 r-
32. q 60
t. P
3 6 I- d
E
g _ . #0
C)
x:
& 1+ :-
P C
‘ II 20
2 h “
D ‘ ‘
51‘9‘40J‘ A , K
0 1 ““'9-?1- 0
ux in U\ tn u\ ‘n E} :9 E} :2 E} :9
a— '- — — — N N N

DAILY AVERAGE VELOCITY INTERVAL, mph

Figure 4.1 Daily Average Wind Velocity pdf's and CDF's.

CUMULATIVE PROBABILITY, %

39

Table 4.2. Length of Harvest Period.

 

 

Variety Days of Harvest
McIntosh 10
Jonathan 14
Golden Delicious 10
Northern Spy 10

 

day of the period. From these results, and the fact that
this is a relatively short period, it was hypothesized

that the mean and standard deviation of daily average
velocity could be assumed constant over the period. Further-
more, using the method discussed in Section 3.2, the cor-
relation between velocities on successive days was esti-
mated at 0.36.

For the purpose of constructing a wind velocity
simulator it was hypothesized that successive velocities
could be described by the same first-order lag model as
discussed in Section 3.2. Random velocities having a
specified mean, standard deviation, and correlation between
successive values were generated using the autoregression

model (16):

r u) + o (l - 02)15 x.

.= + .
WV 11 p(WVl_l 1

l

40

Where:
WVi = Correlated average velocity, day i
WVi-l = Correlated average velocity, day i-l
p = Correlation between WVi and WVi_l
p = Mean of WVi
o = Standard deviation of WVi
Xi = Standardized random variable calculated from:
_ Y-u
X-(o)

Where: Y = Random (uncorrelated) value for
velocity generated from the CDF
for daily average velocity

u,o = Same as above

The model for daily windloss was assumed to be of the form:

WL. — CON , wv. < WVB
1 1
WI.i = CON + D (wvi - WVB)n, wvi 3 WVB
Where:
WLi = Windloss for day i
WVi = Daily average velocity, day i

WVB = Base daily average velocity

CON = Minimum daily windloss (0.002 for McIntosh,
0.001 otherwise)

D = Slope parameter

n = Exponent

41

The annual windloss was assumed to be the ratio of
total volume of windfalls to total volume harvested plus
windfalls. The parameters D, WVB, and n in the daily
windloss model were not available from data, but were
determined using an iterative procedure requiring that the
expected value and pdf for annual windloss, Table 4.1, be
closely matched. This procedure is discussed in the next

Section.

Table 4.3. Wind Velocity Model Parameters.

 

 

9.12 mph 3.26 mph 0.36 mph

 

4.3 Parameter Estimation

 

Table 4.3 gives the grand mean and standard devia-
tion (calculated using residuals from the grand mean for
all observations) for daily average velocity, and the
estimated correlation between successive velocities. The
estimated correlation is based on about 1000 observations
and was assumed to be the correct value. While the
residuals of velocity may be normally distributed, a
significance test of the estimated correlation was not per-

formed because the necessary condition of independence

42

between successive velocity observations is violated by
the hypothesized first-order lag model.

The pdf for velocity was integrated to obtain the
CDF for use in generating Y, the random velocity values.

An iterative program was written for use in esti-
meting the parameters D, WVB, and n. This program used the
annual yield model, wind velocity model, daily windloss
model, lost work time model (discussed in Section 5.2), and
length of harvest period to generate annual windloss
observations for a specific variety over a period of NY
years. The mean and standard deviation of annual windloss
were calculated and the observations were sorted into a
pdf having the same loss intervals as the assumed pdf's
given in Table 4.1.

Preliminary runs were made using exponents, n, of
l, 2, and 3 each with 16 combinations of D and WVB,
covering a narrower range, until the estimated and simulated
pdf's and expected values for annual windloss were in close
agreement, based on 800 years of simulated observations.
The values of D and WVB giving best agreement were used for
all subsequent windloss modeling, and are listed in Table
4.4 along with the expected value, and its standard devia-

tion, of annual windloss.

100 <1

90

80

x

 

K K . ‘ v\ ->

.u. 0 AU 0 Au 0
7 6 5 h- 3 2

.OOO3O2_: u<Ozz< 1O OO.O_O<OOOO

 

Figure 4.2 Response Surface for McIntosh Windloss.

 

540%
15%
20%

WL 3 ID-
I: '5-

 

\vlL

(PIILPIIIKIIIKIIIR. .1

o o O o 0 o
7 6 5 h. 3 2

 

O .OOO3O2_3 3<Ozz< 1O >»_3_O<OO¢O

Figure 4.3 Response Surface for Jonathan Windloss.

5%
SWIUA
lO-I5%
rm. ‘3 '5-20%
0

’\

 

I‘ O>O:.

nu nu nu .u. .u nu AU Au .0 nu nmw
no a, .8 .l ,o c; h. .1 or .I

 

O. .3325 ._<Ozz< 1O $3.28.:

Figure 4.4 Response surface for Golden Delicious Windloss.

46

 

 

 

/hlllK ‘ ( .4 I
0 nu Au 0 .u 0 nu 0
0 a; 1: ,6 .3 h. .1 2

a £8.32; ._<Ozz< 1O E3332:

m = 10-15%
UL - lS-20%

Figure 4.5 Response Surface for Northern Spy Windloss.

47

Table 4.4. Daily Windloss Model Parameters.

 

 

Variety u ou D WVB n
% % mph
McIntosh 12.2 0.28 0.0013 7.0 2
Jonathan 5.9 0.13 0.0007 8.0 2
Golden Delicious 6.0 0.16 0.0010 8.0 2
Northern Spy 6.4 0.16 0.0008 7.5 2

The annual windloss response surfaces were
plotted for each variety, to help visualize the effect of
changes in D and WVB, and are shown in Figures 4.2 through

4.5.

4.4 Model Verification

 

The wind velocity distribution formed by the auto-
regression model was determined by generating 2400 cor-
related velocity values, then sorting them into pdf and
CDF formats. The generated and observed pdf's differed
slightly but the generated and observed CDF's corresponded,
Figure 4.1.

Many assumptions have been made in develOping the
windloss model and in estimating parameters for it. The
only verification offered is that the simulated annual
windloss pdf and expected value agree closely with those

estimated from grower and horticultural agent opinion.

48

The actual relationships which determine annual windloss

are undoubtedly more complex. However, Hillier and
Lieberman (11) state that in constructing a model, "all

that is required is that there be a high correlation between
the prediction by the model and what would actually happen F”-

in the real world." It is thought that this windloss

I'll-C. .

simulator meets their requirement, considering the quality

of data presently available.

 

5. HARVEST RATE--ACREAGE RELATIONSHIP

The number of days available for harvesting is F”

_ '1

determined by fruit maturity requirements and environmental
factors such as rain (which may reduce the days available
for work) and freezing temperatures or snow (which damage

the fruit or terminate harvesting).

 

The harvestrate—-acreageredationship for a mechani-
cal harvest system depends upon: (1) length of the harvest
season; (2) varieties and respective acreage; (3) harvest
rate; (4) accumulated working hours during the harvest
season; and (5) harvest policy of the grower or Operator.
The first four Of these can be determined from available
data or specified as a parameter of the harvest system.

The question of harvest policy was considered as a choice
between two alternatives: (1) begin harvesting each
variety when minimum maturity is reached and continue at a
constant rate until harvest is completed, not exceeding

the date of maximum acceptable maturity; or (2) delay the
start of harvest for each variety by up to 7 days, hOping
for increased yield due to sizing and accepting the risk of
less yield due to windloss, then harvesting at a high rate

so as to complete harvest by the same date as for (l). The

49

50

first policy is referred to as Maximum Acreage, and the

second as Delayed Harvest.

5.1 Available Data

 

5.1.1 Length of the Harvest Season

 

The typical length of the harvest season beginning
with McIntosh and ending with Northern Spy was estimated2
as six weeks with completion about November 1 (to avoid
freezing temperatures and fruit loss). The season may
start early or late, but will have a 6-week length.

For a given season, the early, most likely, and
late starting dates3 for each variety are given in Table
5.1. These correspond to dates recommended for apples
going into controlled atmosphere storage. The typical
length of harvest for each variety was assumed to be the

same as given in Table 4.2.

5.1.2 Varieties and Respective

Acreage

 

The four varieties, in acreage proportions given in

Section 2.8, were assumed as a representative cross-section.

The total acreage was determined by an iterative procedure

which is discussed later in Section 5.2.

 

2Personal communication, Frank Klackle, Dist. Ext.
Hort. Agent, Grand Rapids, Mich.

3Personal communication, Dr. D. H. Dewey, Professor,

Hort. Dept., Mich. State Univ., E. Lansing, Mich.

,,~ .V‘u_ .

 

I”

51

Table 5.1. Harvest Starting Dates, Grand Rapids Area.

 

 

 

Variety . Early Most Likely Late
McIntosh Sept. 10 Sept. 17 Sept. 24
Jonathan ‘ Sept. 20 Sept. 27 Oct. 4
Golden Delicious Oct. 10 Oct. 17 Oct. 24
Northern Spy Oct. 10 Oct. 17 Oct. 24

 

 

illf

5.1.3 Harvester Capacity

 

Harvest rate was considered as a Controllable Input
and Design Parameter in Section 2.5. Harvester capacity,
in total acres per season, was determined by the iterative
procedure (Section 5.2) for several specified harvest

rates.

5.1.4 Accumulated Working Hours

 

Five events were considered to control the number
of working hours per season: (1) length of the harvest
season; (2) defined work week; (3) work time lost due to
rain; (4) work time lost due to machine failure; and
(5) waiting time between completion of one variety and
start of the next.

The length of harvest season has been established
at six weeks. The defined work week was assumed to be six

days and eight hours per day (8 AM-S PM, no Sunday work).

52

The work time between successive machine failures was
(assumed to be exponentially distributed, a common assumption
for well maintained equipment, with a mean occurrence time
of 80 hours.4 Each failure was assumed to cost four hours
of work time or the balance of the workday if less than
four hours remained at failure.4 The waiting time between
varieties was determined by the required completion date
of one variety and the beginning date for the next.

The work time lost due to rain was determined by

the procedure discussed below.

5.1.5 Work Time Lost Due to Rain

 

Hourly precipitation records for the Kent County
Airport, Grand Rapids, were analyzed for the period 1951-
1970. The following criteria5 were used for estimating
work time lost due to rain: (1) a rain of 0.20 inches per
hour or more during the period 5 PM-BAmeould result in
four hours of work time lost during the following 8 AM-

12 Noon period; (2) a rain of 0.03 inches per hour, or

more, during the normal work period would result in that
hour and the remaining hours until 12 Noon (for rain in
morning) or 5 PM (for rain in the afternoon) being lost.

The daily Observations of lost time due to rain were

 

4Based on the author's experience.

5Personal communication, N. D. Strommen, State
Climatologist, Mich. Weather Service, E. Lans., Mich.

53

determined for the 20-year period. The results were in
agreement with grower Opinion in that an average of one-

half day per six-day week will be lost to rain.

5.2 Maximum Acreage Policy

 

A program, which used the information described in
Section 5.1, was designed to estimate the maximum acreage
which could be completed 90 and 95% of the years for
specified harvest rates. The actual lost time Observations
due to rain were used in this analysis and 50 iterations
were made for each of the early, most likely, and late
starting dates. A stochastic model for lost time was
designed later, Section 5.3, when a continued need for
such a model was apparent.

Starting date did not significantly affect the
maximum acreage, assuming a constant season length. The
average results are given in Table 5.2. Coefficient of’
variation for total acres ranged from 1.1 to 1.7%.

The range in number of days between completion of
one variety and beginning of the next is given in Table
5.3. Generally, there were about five idle days per season
due to waiting for the next variety to reach proper
maturity. Personal discussion with two representatives of
large apple processors suggested that any variety could be
harvested five days early, if occasionally necessary, to
improve the harvest schedule. The acreage remaining to be

harvested in the one or two years out of 20 (which

Table 5.2.

54

Harvest Rate--Acreage, Maximum Acreage Policy.

 

Harvest Rate

Total Acres

 

 

 

 

 

 

 

 

 

 

 

 

(trees/hr) Sept. 10 Sept. 17 Sept. 24
.90 .95 .90 .95 .90 (.95
10 69.6 65.6 68.8 65.6 67.1 64.8
15 104.4 98.4 103.2 98.4 100.6 97.3
20 139.1 131.3 137.6 131.2 134.2 129.6
25 173.9 164.1 172.0 164.0 167.7 162.1
30 208.7 196.9 206.4 196.8 201.2 194.5
Table 5.3. Idle Days Between Varieties, Maximum Acreage
, Policy. '
Idle Days
Varieties
(finished/ Sept. 10 ...Sept..l7 . ...Sept-.24
started)
.90 .95 .90 .95 .90 .95
McIntosh/
Jonathan 1-4 0-4 0-5 1-5 1-5 1-5
Jonathan/
G. Delicious 0-5 1-5 0-5 1-6 0-5 1-6
G. Delicious/
N. Spy 0 0 0 0 0 0

 

55

determined the .95 and .90 probability level of completion)
equalled about one full day of operation. Thus, if
harvest was started earlier (when necessary) and two days
were added to the length of the harvest season the maximum
acreage values might increase by 10%. However, the cal-

culated values will be assumed to hold.

5.3 Delayed Harvest Policy

 

A second program was designed for use in estimating
the expected change in harvested volume, annual windloss
volume, and associated risk levels faced by a grower
considering the Delayed Harvest Policy.

The McIntosh variety was examined first. A 40-acre
planting, at 27 trees per acre, and a harvest rate of 15
trees per hour were assumed in order to establish a
nominal harvest period of 10 days. Eight identical harvest
systems were assumed to begin harvesting on day 1, day 2,
... day 8, each with a harvest rate sufficient to complete
the 40 acres by the same date. For purposes of simplicity,
machine recovery was assumed at 100%, fruit injury at 0%,
and machine breakdowns at zero. The yield model, windloss
model, and daily growth rate already discussed were used
in this program. In addition, stochastic models for lost‘
time due to the occurrence of a Sunday or rain were used.

The model for lost time due to a Sunday consisted of

comparing a uniform (0, 1) random number to the daily

56

probability of occurrence of a Sunday. The successive daily
probabilities were 1/7, 1/6, 1/5, 1/4, 1/3, 1/2, and 1/1.
The first Sunday occurred when the random number was less
than the daily probability. Succeeding Sundays were then
specified at 7-day intervals.

The model for lost time due to rain used a discrete
pdf for lost time, Table 5.4, developed from the rain data

discussed in Section 5.1.5. Using the method discussed in

 

Section 3.2, the correlation between lost time for successive
days was concluded to be zero. Thus, although rain is
typically assumed to be a persistent phenomena (i.e., amount
on successive days is correlated), lost work time is not a
persistent phenomena (for the assumed criteria, Section
5.1.5). Lost time due to rain was generated daily using

the inverse transform method. Work time available was

equal to eight hours minus the lost time.

Table 5.4. Daily Work Time Lost Due to Rain.

 

Hbunslost 0 1 2 3 4 5 6 X7 8

 

Prdoability 0.816 0.007 0.008 0.014 0.107 0.006 0.011 0.012 0.018

 

57

The delayed harvest simulation was run for 200
years, summary statistics were calculated, and the annual
observations for each harvest policy were sorted into pdf
format. The results are summarized in Table 5.5.

The simulation results for McIntosh, at the average
growth rate, show that: (1) the harvest ratio (ratio of
volume harvested with delay to volume harvested without
delay) decreases with length of delay; (2) the mean annual
windloss increases with length Of delay. However, for
growth rates 1 and 2 standard deviations greater than the
average: (1) the change in harvest ratio is insignificant;
(2) the mean annual windloss increases with length of
delay. In addition to these conclusions, the standard
deviations for both harvest ratio and windloss suggest that
a delayed harvest policy is very risky for McIntosh.

The Jonathan variety has about one-half the tendency
for windloss as does the McIntosh and has a longer harvest
period. A second simulation was run using data for
Jonathan and performing an identical analysis. The
results are summarized in Table 5.6.

In general, the conclusions for McIntosh also apply
to Jonathan. The decrease in harvest ratio and increase
in windloss for the average growth rate are not as great
as for McIntosh. The results for growth rates 1 and 2
standard deviations greater than the average both result

in a slight increase in harvest ratio. The average

wt“!

 

“I.

58

 

 

 

 

 

Table 5.5. Delayed Harvest Results, McIntosh.
Harvest Policy Harvest Ratio Annual Windloss
Delay Rate 0 o H OH
Days Trees/hr % %
3 0 15.00 1.000 f -- 11.3 0.50
g 1 16.88 0.992 0.0096 12.3 0.55
g 2 19.29 0.988 0.0187 12.8 0.57
g 3 22.50 0.983 0.0280 13.5 0.61
w 4 27.00 0.977 0.0356 14.2 0.63
g 5 33.75 0.971 0.0426 15.0 0.66
3 6 45.00 0.965 0.0516 15.8 0.69
E 7 67.50 0.966 0.0517 15.8 0.69
g 0 15.00 1.000 -- 11.1 0.49
m,U 1 16.88 0.999 0.0088 12.1 0.54
3:88 2 19.29 0.998 0.0169 12.6 0.56
353;; 3 22.50 0.997 0.0256 13.2 0.59
38.5 4 27.00 0.996 0.0334 13.9 0.62
3513 5 33.75 0.996 0.0410 14.5 0.64
§‘* 6 45.00 0.995 0.0506 15.3 0.67
< 7 67.50 0.996 0.0516 15.2 0.67
3 0 15.00 1.000 -- 11.0 0.48
End“) 1 16.88 1.005 0.0087 11.9 0.52
5.35 2 19.29 1.008 0.0170 12.3 0.55
Egg» 3 22.50 1.012 0.0253 12.9 0.58
”8'; 4 27.00 1.015 0.0332 13.5 0.60
§OIB 5 33.75 1.019 0.0411 14.1 0.62
34' 6 45.00 1.024 0.0516 14.8 0.65
5 7 67.50 1.025 0.0522 0.65

14.7

 

 

 

 

 

 

 

Table 5.6. Delayed Harvest Results, Jonathan.
Harvest Policy Harvest Ratio Annual Windloss
Delay Rate u o u ou
Days Trees/hr % %

3 0 15.00 1.000 -- . 0.28
8 1 16.36 0.999 0.0038 0.29
g 2 18.00 0.997 0.0076 0.31
g 3 20.00 0.996 0.0103 0.32
0 4 22.50 0.995 0.0134 0.34
§ 5 25.71 0.994 0.0147 . 0.34
g 6 30.00 0.992 0.0174 . 0.36
E 7 36.00 0.993 0.0182 7. 0.36
3 0 15.00 1.000 -— . 0.27
g 1 16.36 1.003 0.0037 . 0.29
35%;: 2 18.00 1.004 0.0074 . 0.30
82.2 3 20.00 1.006 0.0099 . 0.31
Egg 4 22.50 1.007 0.0128 . 0.33
3,.5 5 25.71 1.010 0.0141 . 0.33
§+-Q 6 30.00 1.012 0.0168 0.34
g 7 36.00 1.011 0.0174 . 0.35
3 0 15.00 1.000 -- 5. 0.27
8,: 1 16.36 1.007 0.0043 5. 0.28
fig: 2 18.00 1.011 0.0091 5. 0.29
Egg-E 3 20.00 1.015 0.0114 6. 0.31
ӣ33. 4 22.50 1.020 0.0144 . 0.32
§°¢§ 5 25.71 1.025 0.0160 . 0.32
§'* 6 30.00. 1.031 0.0187 6.8 0.33
s 7 36.00 1.029 0.0176 . 0.34

.......................

 

 

‘IIE

60

results and standard deviations suggest that a delayed
harvest policy for Jonathan is less risky than for
McIntosh but, the expected increase in harvest ratio is
not of sufficient magnitude to be economically attractive
to commercial growers.

From a positive viewpoint, these results also
suggest that if a delay (due to manpower, scheduling,
machinery, etc. problems) should occur, and the harvester
is capable of a higher harvest rate, the harvested volume

will not decrease greatly.

5.4 Harvest Policy Selection

 

After considering the results of the Maximum
Acreage and Delayed Harvest Policies, the former was
(selected for use in the final harvest simulation model.

. Several intangible factors also supported the
selection of this policy: (1) a harvest policy should not
deliberately expose the grower to an increased risk of
substantial financial loss; (2) high growth rates are
required for a delayed harvest policy to have a low risk
of financial loss, but only an indication of growth rate
is available at the actual start of harvest; (3) many of
the harvest rates required in the Delayed Harvest Policy
are well in exceSs of those attainable in the near future

on standard type trees; (4) the material handling problem

61

(scheduling of bulk bin delivery and pickup) becomes
critical, by present standards, when the harvest period

is compressed; and (5) the effect of machine failure (not

included in the delayed harvest simulation) becomes critical

as the harvest period is compressed.

 

 

rW11..-.. £5.

6. HARVEST SIMULATION MODEL

The program HARVSIM uses all inputs and models
discussed in previous chapters to simulate both hand and
mechanical harvesting of apples for processed utilization.
HARVSIM is written in FORTRAN IV and requires about 22,000
octal units of central memory to compile and run on the
CDC 6500 Computer.

A simplified flow-chart of HARVSIM is given in
Figure 6.1 to show the simulation sequence and subroutines
used. The entire program is described in "HARVSIM--
Evaluation of mechanical harvest methods for processing
apples via simulation" which was submitted for publication
in the MSU, Agricultural Engineering Department Preprint

Series.

6.1 Initialization

 

The number of simulations completed before the
output statistics are calculated and printed is controlled
by NG, i.e., the number of growers assumed in the sample
which the statistics represent. Any number can be set
without changing the input required or output format used.

The basic dimensions of the simulation are con-

trolled by: (1) NY, the period of consecutive years

62

63

““3 i) g (P

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 
  

 

   

 

COMMON . A 00 65 [ CALL HARVEST
DIMENSION JY=I,NY
Specify [I CALL WIND ] Variety
NG, NY, NH, NVAR ' ' Finished

 

 

 

 

 

 

 

[ Generate Random]

Initialize Number Array
Yield cor * .~Lc0NTINUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

JVAR=I
I NEXT=JVAR+I

Read ' — .{ CONTINUE
, _ ,

CrOp, Harvester,

and Labor Params. @—[JDAY=JDAY+I - I _
~ I CALI. FINAL J

wind cor
I 0° 7° 3
JH=2,NH wONTINUE
Test and Print ' i
Yield cor ' .
wind CDF l JDAY=0 J CALL GROSUM j 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Initialize I
KS, RP, ROT Print
0 Summary
Print (:g ’ ::)
Crop, Harvester, END
and Labor Params. CALL GROWTH
' .
Set CALL WINDLOSS j o
Accumulators-0.0 ’
CALL WORKS J Check
[ 00 60 - Sunday
- JG-I,NG .Schedule -

 

 

 

 

[ CALL YIELD j

6

Figure 6.1 Simplified Flow Chart of HARVSIM.

 

 

64

analyzed for each grower; (2) NH, the number of harvest
methods analyzed in the simulation; and (3) NVAR, the
number of varieties considered in each simulation. Changes
in NY or NVAR will require corresponding changes in input
data and output format used. A change in NH requires a
corresponding change in input data, but no change in output
format.‘

The CDF for annual yield of each variety and for
daily average wind velocity are each initialized using
yield or velocity values and the corresponding cummulative
probability values. Function TABLI (16), a FORDYN program,
computes the uncorrelated yield and velocity values based
on the initialized CDF's. Initialization is checked by
printing 21 points of each CDF (from 0.0 to 1.0) and the
corresponding yield or velocity values, using function
TABLI .

The crop parameters, harvest parameters, and labor
costs are read in from three groups Of data cards. The
most likely number of days between start of harvest for
the successive varieties are established with the
variables KS(JVAR). The typical decay and recovery of
windfall apples are established for each variety using
ROT(JVAR) and RP(JH). The initialized values for the three
categories of parameters are then printed for a check on

initialization and future reference.

 

65

6.2 Subroutine YIELD

 

A new set of correlated yields for each variety
are generated for NY years, for each grower, and stored

in a yield array.

6.3 Subroutine WIND

 

A new set of correlated daily average wind
velocities are generated for each year and stored in a

wind velocity array.

6.4 Random Number Array

 

Separate series of consecutive uncorrelated random
numbers (0,1) are generated and stored for use in deter-
mining occurrence Of the first Sunday, lost time due to
rain, and machine breakdowns. By storing these numbers
at this time, each harvest method can work through the
harvest season on the same schedule, have the same daily

rain occurrences, and the same failure times.

6.5 Subroutine GROWTH’

 

Growth of on-tree yield is assumed to occur at an
average rate between each successive day after maturity
is reached. If the next variety is mature, but is not yet
(being harvested, the appropriate growth rate is also

applied to it.

66

6.6 Subroutine WINDLOSS

 

Windloss of on-tree yield is assumed to occur on
each day after maturity is reached. If the next variety is
mature, but is not yet being harvested, windloss is also
calculated for it. Windloss is a function of wind velocity r‘
each day and the windloss-velocity relation for the

respective variety. This subroutine first calculates the

 

windloss (percentage loss of on-tree yield each day), then

calculates the actual on-tree yield and on-ground yield

 

for day i.

6.7 Subroutine WORK

 

The harvest time available each day depends on:
(1) if the day is a Sunday or workday; (2) if work time is
lost due to rain; and (3) if a machine breakdown occurs.
Using the previously generated random numbers, these
events are determined daily by this subroutine and a value

for HT (0.0 3 HT 1 8.0) is calculated.

6.8 Subroutine HARVEST

 

If HT is greater than zero, harvest proceeds. The
volumes harvested, remaining trees, and hours of operation
are accumulated, and control returns to the main program
where JDAY advances by l and the process is repeated.

When one variety is finished, the next is started on the

same day if it is mature. When all varieties are completed,

67

HARVEST instructs the main program to advance to the next

harvest method and the season is repeated.

6.9 Subroutine GROSUM

 

When all specified harvest methods and years have
been completed for one grower, this subroutine calculates

the sums and sums-of-squares of the output variables for

 

later use in calculating expected values and standard

deviations.

6.10 Subroutine FINAL

 

When the simulations have been completed for all
growers, the expected value and standard deviation of each

output variable are calculated.

6.11 Output

 

The results calculated in FINAL are given in
tabular form with appropriate labelling and footnoted
explanations.

Summaries, rather than yearly values for the output
variables, are given for ease of analysis. The results
are determined by a combination of several random processes.
The actual observations can be used to estimate the pdf
for each variable, which in turn can be used to estimate
the probability of occurrence of certain values for each

variable.

 

7 . HARVEST SIMULATION

Input information required for HARVSIM was used to
simulate results for growers with 70 acres of mature,
standard type, apple trees (the four varieties in propor-

tions given in Section 2.8) and two sizes of mechanical

 

harvest systems. The expected value and standard devia- fl
tion were calculated for each output variable listed in

Section 2.4. Histograms for each output variable were

inspected for reasonable similarity with the Normal Distri-

bution, and the probability of negative margin and planning

margin were estimated. Finally, to determine the effect

of input parameter changes on output variable values, a

sensitivity analysis was conducted.

7.1 Simulation Assumptions

 

The two mechanical harvest systems were assumed
to be self-propelled, and commercially available in 1968
at $15,700 for a 10 tree per hour system and $19,600 for
a 15 tree per hour system. The equipment costs are thought
to be representative estimates for each harvest system
in 1968.

The mechanical harvesting of apples is not an
established practice in any part of Michigan. During the
first few years of transition from hand harvest methods to

68

69

mechanical harvest methods, machine obsolescence will
occur in a short period of time. An even number of years
should be used for the planning period because annual
yields are biennial, as discussed in Chapter 3. In
addition, the simulation results will be more representa-
tive if the most recent input data available is used.
Because of these facts, a four-year machine life and
planning period, 1968-1971, was assumed.

The yield for each variety, generated by subroutine
YIELD, was assumed to be independent of the yield for the
other varieties. This assumption was made because the
four varieties being used represent a cross-section of the
varieties a grower may have, and enough accurate data is
not available from which correlations between varieties can
be statistically confirmed. An assumption that yield for
each variety will vary in the same pattern would result
in higher standard deviations, but the same expected values,
for margin and planning margin.

Estimates for the portion of yield: (1) Zk-inch
diameter and larger; and (2) with natural defects, were
obtained for fruit of each variety.6

Estimates for the portion of yield: (1) removed

from the tree; and (2) with excessive harvest injury,

 

6Personal communication, Frank Klackle, Dist.
Ext. Hort. Agent., Grand Rapids, Mich.

70

for each variety were also obtained.7 The estimates for
each mechanical harvest system are optimistic, i.e., the
best results anticipated.

The volume of apples recovered as windfalls, and
the ratio of windfalls for the mechanical harvest methods
compared to the hand harvest method were computed. However,
the cost of recovering windfalls and the value of windfalls
were not used in determining margin and planning margin
because most growers do not have a dependable market for
this class of fruit.

The piece rate for hand picking given by Kelsey,
Harsh, and Belter (15) was assumed for 1970. Costs for
other years were estimated by assuming a 5% increase
between successive years.

The hourly cost for labor was assumed at a higher
level than nOrmally paid for agricultural labor. Crews
Operating mechanical harvesters are often paid on an
incentive plan to encourage high productivity and dis-
courage careless Operation or practices which decrease
fruit quality. Also, if wages comparable to those in
manufacturing could be paid for agricultural labor, worker
earnings and the supply of labor for agricultural work

would both increase. For these reasons the hourly cost

 

7Personal communication, J. H. Levin, Invest. Ldr.,
Fruit and Veg. Harv. Invest., USDA-ARS-AERD, E. Lans.,
Mich.

 

71

of labor was assumed comparable to the average hourly wage
in manufacturing for the State of Michigan (20).

All assumed crop, price, and labor cost parameters
are given in Table 7.1, and harvest system parameters are
given in Table 7.2, under the apprOpriate name as described W“
in Sections 2.2 and 2.3. I

Accurate outputs from HARVSIM depend on: (1) accu-
rate input information; (2) valid model formulation and

accurate parameter estimation; and (3) the number of

 

growers (i.e., number of independent observations)
included in the simulation. Of the output variables given
in Section 2.4, the most important is planning period
margin. The number of growers to include in the simulation
was determined by using HARVSIM results for 80 growers to ’
estimate the standard deviation of: (l) the mean of
expected planning margin (EPMAR); and (2) the standard
deviation of planning margin (SDPMAR). The change in
accuracy of estimating EPMAR and SDPMAR with a change in
the number of growers was then considered. The above
standard deviation estimates were determined by the

relations:

Where:

72

 

 

 

 

 

 

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mo. mo. mo. mo. Hm

 

73

 

 

 

 

mm. mm. mm. mm. mm AOL HOOHGOQOOZ
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.mumqumHmm Ewummm umo>Hmm pwEdmmd .N.h OHQMB

74

S? = Standard deviation of the mean
S = Standard deviation of the observations

Number of observations

:3
II

 

 

and '
S

S’s-775% ,~

Where: 5
SS = Standard deviation of S t

The relation of S; is valid for samples from any popula-
tion with finite variance and the relation for SS holds
for samples of n 1 15 from a population considered to be
normally distributed (26). Planning margin can be con-
sidered normally distributed as discussed in Section 7.2.

For 80 growers the standard deviation of the mean
of EPMAR was about 195, or 3-7% of EPMAR depending on the
harvest system, and the standard deviation of SDPMAR was
about 140, or 8% of SDPMAR. Because both standard devia-
tion values vary inversely with the square root of the
number of observations, corresponding standard deviation
values for 40 growers should be about 1.4 times greater
than for 80 growers. Similarly, values for 80 growers
should be about 1.4 times greater than for 160 growers.
The accuracy obtained by including 80 growers in the

simulation was considered adequate.

75

7.2 Simulation Results

 

The expected values and standard deviations for
the output variables are given in Tables 7.3 and 7.4.
Harvester number 3 is the $15,700 system with a harvest
rate of 10 trees per hour and harvester number 4 is the
$19,600 system with a harvest rate of 15 trees per hour.
The results in Table 7.3 show that:

1. Expected margin increased steadily since 1968 (the

H.433 “haul .‘ '

only year in which expected margin was negative for

 

‘l' -

either harvest system).

2. Expected planning margin was positive for both harvest
systems but system 4 was $4,300 higher than system 3.
The increased margin was due to the assumed wage rates
and annual savings in man-hours for harvesting with
system 4 compared to system 3.

3. If successive yields were independent, the standard
deviation of planning margin would be equal to the
square root of the sum of the squared standard devia-
tions of margin, Table 7.3. However, the negative
correlation used in the yield model caused the standard
deviation of planning margin to be 36% less than would
result if successive yields were independent (planning
margin is the sum of four correlated margins). Thus,
when expected planning margin is positive, the proba-

bility of negative planning margin is less than would

result if successive yields were independent.

 

   

76

 

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77

 

 

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II HOOHumHumum Houmo>umm
wuwHHm> mHmmd

 

.moHumm OEDHO> was >UH>Huospoum OmumHOEHm .v.h OHQOB

78

4. Overall cost per bushel for mechanical harvesting
remained relatively constant over the four-year period.

The results in Table 7.4 show that:

1. Worker productivity for system 3 was about three times
higher, and for system 4 was about five times higher,
than for hand picking assuming an expected productivity
of 10 fOr hand picking (23).

2. The standard deviation of productivity was 22-38% of
the expected value, depending on the variety and
harvest system.

3. The harvested volume ratios for system 3 indicate
that 3-6% less volume per grower, depending on variety,
would be available for processing or juicing by pro-
cessors. Similar ratios for system 4 indicate that 2-4%
less volume would be available.

4. The harvested volume ratio for windfalls for system 3
indicates that 68-1l6% more volume would be available
for sale (at the assumed recoveries for hand and
mechanical harvest) with mechanical harvesting. The
increased volume is a combined result due to equipment
breakdown and higher recovery of windfalls with a
mechanized system. No shortage of labor was assumed
for hand harvesting but recovery of windfalls is
usually low because windfalls are picked up by hand if

a readY’market exists after hand picking is completed.

 

79

5. The harvested volume ratio for windfalls for system 4
indicates 11% more to 35% less volume available for
sale with mechanical harvesting. This occurs because
system 4 has a higher harvest rate than needed by the
assumed 70-acre grower, thus each variety is harvested
in a shorter time period than required for hand harvest.

The probability that margin (YMAR) or planning
margin (PMAR), for an individual grower, may be less than
or greater than a particular value can be estimated using
the results in Table 7.3, if the respective YMAR or PMAR
pdf is known. Since each grower in the sample is inde-
pendent of all other growers, and the correlation between
yield for all varieties each year iszero, the yearly YMAR
observations and the summary PMAR observations are
independently distributed about their expected values. The
pdf of each output variable was estimated by sorting the
individual observations into a frequency histogram which
was centered on the expected value and divided into six
parts of one standard deviation each. By inspection of the
histograms it was concluded that YMAR, PMAR, and productivity

(PROD) could be assumed to follow the Normal Distribution.

However, the histograms for harvest cost (HC) and windfall

ratio (DR) were skewed to the right of the mean and those

for processing ratio (PR) and juice ratio (JR) were skewed
to the left of the mean. Thus, these latter random

variables are probably not normally distributed.

.’1 .

 

II .’. 'v‘ - -

80

The probability of negative YMAR and PMAR was
estimated using the assumption that YMAR and PMAR are
normally distributed. The results for harvest system 3,
Table 7.5, show that the risk of negative YMAR steadily

declined from 0.74, the first year of the planning period, r=

In! N '

to 0.19, the last year of the planning period, and that
the risk of negative PMAR is only 0.05. The trend is

similar for harvest system 4 but the corresponding risks

 

are less than one-half those for harvest system 3.

[I

Table 7.5. Risk of Negative Margin.

 

 

 

Harvester YMAR PMAR
Number
1968 1969 1970 1971 1968-71
3 0.74 0.32 0.21 0.19 0.05
4 0.38 0.10 0.10 0.02 0.00+

 

7.3 Sensitivity Analysis

 

A sensitivity analysis can provide: (1) greater
insight to the inner workings of the simulation model;
(2) an identification of the critical and less critical
parameters; (3) an indication whether some of the con-
straints should be loosened or tightened; and (4) a more
quantitative idea about the expected overall performance

of the system being modeled (1).

81

Twelve parameters were selected for inclusion in a
sensitivity analysis on expected margin and expected plan-
ning margin performed for both mechanical harvest systems.
One parameter at a time was varied and the corresponding
outputs were calculated by HARVSIM. To reduce computer
time, but still obtain acceptably accurate expected values,
only 40 growers were included. Each complete simulation was
made using the same series of random numbers, so the dif-
ference in output was due only to the harvest system and
the particular combination of parameter values. The
initial parameter values were the same as used in the pre-
viously discussed simulation for 80 growers, Tables 7.1
and 7.2.

The sensitivity results are given in Figures 7.1-
7.12 for harvest system 3, and Figures 7.13-7.24 for harvest
system 4. The initial parameter values are indicated by
the symbol A along the horizontal axis of each figure.
Variation in parameter value is given in absolute value,
factor value, or delta value (A) depending on the parameter.
Factor values are simply 0.75, 1.00, or 1.25 times the
initial absolute value of the parameter. Delta values are
a constant difference from the initial absolute values and
were used when the initial absolute value was different
for each variety, quality class, or year. The sensitivity
to parameter variation is indicated by dashed lines for
expected margin (EYMAR) and by a solid line for expected

planning margin (EPMAR).

EPMAR a EYMAR, $1000

EPMAR 5 EYMAR, $1000

IO

10

82

 

 

 

Figure 7.1 Sensitivity to Equipment Cost.

Uj—‘U

U 1

 

 

Figure 7.2 Sensitivity to Crew Size.

EPMAR s EYMAR, $1000

EPMAR 5 EYMAR, $1000 .

EPMAR & EYMAR, $1000

12

IYYUU'IUIjIU‘

V

 

 

 

HRATE

Figure 7.3 Sensitivity to Harvest Rate.

 

 

O
TITIV1TfiIIIIv

 

Figure 7.4 Sensitivity to Machine Recovery.

 

 

Figure 7.5 Sensitivity to Fruit Injury.

EPMAR a EYMAR, $1000

EPMAR s EYMAR, $1000

 

84

 

 

 

Figure 7.6 Sensitivity to Total Acreage.

 

 

 

Figure 7.7 Sensitivity to Yield.

 

EPMAR a EYMAR, $1000

EPMAR 5 EYMAR, $1000

EPMAR & EYMAR, $1000

-5

85

 

 

 

' ____________ ‘r-l97l
- :::::::::::"-197o
____J\ :_ ____‘__ 1 ‘K4969
*’ -.IO -,05 0‘1968
b ‘39

Figure 7.8 Sensitivity to Fruit Size.

 

 

 

r
L- 7

. 1971
r-—-\$\—-‘e; ______ 52::fiaru-ru-J.~4968
’ 0 +u05 +u10

1- ADNAT

 

 

W _ ‘ ‘ — —-_-..a-—1969

*- -.5 ——'o-—_____fl.00
‘-1968

. APROCESS PRICE

. --25 0 + .50

- £3 JUICE PRICE

 

Figure 7.10 Sensitivity to Fruit Price.

 

 

S\000

EPMAR 5 EYMAR ,

S\000

£911». 0 EV MAR,

86

\0
S
‘97\-'\
\970 =-_.______.__
\968-“__ — -—__'—== _
0 "' - —— _=-
—2 00 —100 0 + 00—_'2‘.00

\5
10
5
/
/ /
/// /
¢// /
0 / / /
‘97\-\—_;\0¢/ 0 +.10
‘970~////{PCOST
\968d/
5

 

87

The sensitivity analysis for harvest system 3

shows that:

l.

The relation between EYMAR or EPMAR and all parameters
is linear, with the exception of harvest rate, Figure
7.3, which has a positive curvilinear relation.

EYMAR and EPMAR are sensitive to variation in all
parameters, but are least sensitive to variation in
fruit size, Figure 7.8, and natural defects, Figure
7.9. Input information on fruit size and natural
defects need not be described stochastiCally or
extremely accurate for simulation purposes unless
actual variation is greater than presently assumed.
The magnitude of EYMAR increased each successive year,
but its slope remained nearly constant.

EYMAR and EPMAR increase as fruit price decreases,
Figure 7.10. The large change in fruit price between
1968 and 1970 accounts for the large change in EYMAR
between 1968 and 1970. Low fruit price, typical of
1970 and 1971, favors use of mechanical harvest
systems.

A relatively small decrease in machine recovery,
Figure 7.4, or increase in fruit injury, Figure 7.5,
would result in a negative EPMAR. The magnitude of
EYMAR increased from 1968 to 1971 so that less machine

recovery or more fruit injury could be allowed before

EYMAR would become negative.

 

 

10.

ll.

88

The sensitivity to variation in machine recovery
indicates that orchard modification (for mechanical
harvesting) which would reduce yield an amount equiva-
lent to a .10 decrease in machine recovery would
seriously reduce the expected margins.

EYMAR and EPMAR can be substantially increased by an
increase in harvest rate from 10 to 15 trees per hour,
Figure 7.3. This indicates that orchard design,
machine design, and machine management should be aimed
at achieving a harvest rate of more than 10 trees per
hour. About 60% of the increase in EPMAR from 10 to
15 trees per hour can be accounted for in labor
savings.

EPMAR was negative for about 60 acres or less, Figure
7.6, but the trend of EYMAR indicates mechanical
harvesting is becoming feasible for smaller acreager
growers.

EYMAR and EPMAR were highly positive for growers with
25% higher average yields than assumed in developing
the yield simulator, Figure 7.7.

EPMAR will increase about $3500.00 for each $1.00
reduction in assumed hourly labor cost, Figure 7.11.
EPMAR will increase about $12,000.00 for a $0.10 per
bu increase in the piece rate paid for hand picking,

Figure 7.12.

The

to

89

sensitivity analysis for harvest system 4 shows:

The same types of relations and trends as for harvest
system 3.

Less sensitivity to variations in the number of
Operators per system, Figure 7.14; the hourly cost of F -
labor, Figure_7.23; and fruit price, Figure 7.22, than %
harvest system 3.

The same sensitivity to variations in machine recovery,

Figure 7.16; fruit injury, Figure 7.17; fruit size,

 

IT‘

Figure 7.18; natural defects, Figure 7.19; and the
piece rate paid for hand picking, Figure 7.24, than
harvest system 3.

More sensitivity to variations in equipment cost,
Figure 7.13; total acres, Figure 7.18; and yield,
Figure 7.19; than harvest system 3.

The sensitivity and magnitude of EYMAR and EPMAR

each parameter depends on the initial combination of

parameters selected. However, the initial parameter

values and these sensitivity results are thought to be

realistic.

7.4 Harvest Simulation Conclusions

 

1. Sufficiently accurate estimates of the expected
value and standard deviation of planning margin for
a mechanical harvest system can be obtained from

HARVSIM by including 80 growers in the simulation.

EPMAR a EYMAR, $1000

EPMAR s EYMAR, $1000

0

U1

90

    

 

 

, E E = 2 \

- ~i “:Es=i 1%

_ -. ...__ ‘~ ~.::;EE ==jfi397o

—'\I\ ‘ ‘\ \ ¥‘ 3“”59
0.75 1.0 ‘Q35/1968

t ECOST FACTOR

Figure 7.13 Sensitivity to Equipment Cost.

 

Figure 7.14 Sensitivity to Crew Size.

0—
‘
—
—

1m
1 HI

I97]
:=:ar-l970
"“*~1969

EPMAR 5 EYMAR, $1000

EPMAR 5 EYMAR, $1000

91

  
  

 

 

 

 

 

 

 

EPMAR & EYMAR, $1000

12 -
9 1-
6 _
3 r ar—I97l
0' z: “-1968
C. 10 " 15 20
rmATE
Figure 7.15 Sensitivity to Harvest Rate.
6 1-
} ’K’l97l
3 . ._—a,—4970
’ —- —-~—-1969
L. n“
0 ‘\l968
-3 p
I
Figure 7.16 Sensitivity to Machine Recovery.
)-
6 1-
3 r
#- -
o W
L
b
-3L

 

 

Figure 7.17 Sensitivity to Fruit Injury.

EPMAR a EYMAR, $1000

EPMAR s EYMAR, $1000

92

U1

U j I V 1

    
 

1

 

 

- ' TOTAL ACRES

Figure 7.18 Sensitivity to Total Acreage.

 

 

-5.

Figure 7.19 Sensitivity to Yield.

‘,.1971

,,. ::: :::::: -—- "’ ‘*~1969
0 +12, *- . i... -— —- ‘1"958

 

 

EPMAR : EYMAR, $1000

EPMAR a EYMAR, $1000

EPHAR 5 EYMAR, $1000

93

 

U I T I

 

 

" ________________’-l97l

. :::::::::::*l970

_ “\1969
f"" """ “":"""' *' ""f;f-1968

P ‘J -.10 -.05 o

L cspc

Figure 7.20 Sensitivity to Fruit Size.

 

 

" ____________ r1971
1. ::::::::::::\1970
. _ __ _____\1969
__ F._A_ — — - "' ‘1— ,\1968
_ 0 +.05 -.Io

.. ADNAT

 

 

:: ::: —————————— ""97'
. _. ...... :: :::::: :: ::: :::::::::::"~1970
_ _ _ __ __ \1969
, :fi ——————— 3‘1968
.. J -..50 0 +1.00
1. APROCESS PRICE .
.. - -.25 o + .50
. AJUICE PRICE

Figure 7.22 Sensitivity to Fruit Price.

 

94

 

0 +1.00 — +2.00
liOCng

   

Sensitivity to Labor Cost.

\5

 

$1000
‘6

EPHAR 5 EYMAR,

 

95

The expected value of margin increased steadily
during the assumed planning period 1968-1971.

The expected value for planning margin was positive
for both assumed mechanical harvest systems.

The probability that an individual grower will
experience negative margin or negative planning
margin, or any other range of margins, can be
estimated using the Normal Distribution and the
simulation results for expected margin, standard
deviation of margin, expected planning margin, and
standard deviation of planning margin.

The probability that an individual grower had a
negative planning margin was estimated at 0.05

for harvest system 3 and 0.00+ for harvest system 4.
The simulation and probability results for planning
margin indicate that the use of either mechanical
harvest system was feasible during the 1968-1971
period.

The sensitivity analysis shows that expected
margin and expected planning margin are sensitive
to all 12 parameters included in the analysis,
being least sensitive to variations in fruit size
and natural defects.

The sensitivity analysis shows that machine recovery
and fruit injury are critical harvest system

design parameters which, with only small undesirable

96

changes, can result in negative margins. Harvest
rate is another important parameter in terms of
its effect on margins.

9. The expected planning margin is highly sensitive

to the piece rate paid for hand picking, increasing

about $12,000.00 for a $0.10 increase in the

piece rate.

Many additional conclusions can be drawn from the
sensitivity analysis regarding tradeoffs, desirable
harvester features, influence of wage rates, etc.,

depending on one's point—of—view.

 

Ii.

8. APPLICATIONS OF SIMULATION RESULTS

The information generated by HARVSIM can be of
maximum benefit if it is used to guide research planning,
harvest system design, and the management of mechanical
harvest systems.

For research planning the results generated by
HARVSIM, Tables 7.3 and 7.4, the probabilities of negative
margin (YMAR) and planning margin (PMAR), Table 7.5, and
the sensitivity analysis are adequate for identifying
feasible harvest system proposals and research areas which
need initial emphasis.

A thorough understanding of the sensitivity analysis
results will be beneficial for harvest system management,
particularly when YMAR and PMAR are highly sensitive to
changes in machine recovery, fruit injury, harvest rate,
and hand picking cost.

All the above information is useful for harvest
system design. However, a formalized procedure for using
simulation results to select design parameter values does
not presently exist. A selection criteria and procedure

are proposed in the following Sections.

97

 

98

8.1 Criteria for Selecting Design
’Parameter Values

 

 

As stated in Section 1.2, present techniques for
selecting the necessary or allowable values for design
parameters involve computing breakeven conditions, for
harvest costs or grower returns, between the conventional
hand harvest method and the proposed mechanical harvest
method. But, because YMAR and PMAR are normally distributed,
this procedure results in parameter values which cause 50%
of the growers to have negative YMAR (or PMAR) for the
assumed conditions. The standard deviations for YMAR and
PMAR, given in Table 7.3, suggest that YMAR and PMAR will
be very negative for some growers, which is clearly
undesirable.

An alternate criteria to consider is one which
requires a low probability of negative PMAR. If this
probability was set at 5%, parameter values would be
selected so that 95% of the growers would have a positive
PMAR. Table 7.5 shows that harvest system 3 would have
met this criteria and harvest system 4 would have exceeded
this criteria. However, the yearly results for harvest
system 3 show that 74% of the growers would have experi-
enced negative YMAR in 1968. Thus, in addition to a low
probability of negative PMAR, it is necessary that allow

probability of negative YMAR be achieved also.

 

99

8.2 Probability-—Parameter Relations

To select parameter values based on a low proba-
bility of negative YMAR and PMAR, the relations between
probability and various combinations of parameter values
must be available. The results and conclusions in Chapter
7 suggest that these relations can be conveniently
described in graphical form.

In Section 7.2 YMAR and PMAR were assumed to be
normally distributed random variables. The relation
between such variables and their cummulative probability
of occurrence will plot as a straight line on normal
probability paper (26). The sensitivity analysis shows
that both expected margin (EYMAR) and expected planning
margin (EPMAR) are related linearily to 11 of the 12
parameters. For harvest system 3, EPMAR will increase
about $860.00 for each 5% decrease in equipment cost,
Figure 7.1. Using this EPMAR--equipment cost relation and
the standard deviation given in Table 7.3 the probability
of negative PMAR can be determined for various equipment
costs. If the relation between equipment cost and
probability of negative PMAR is plotted on probability
paper a straight line will result because: (1) EPMAR
and equipment cost are linearily related; and (2) PMAR

is normally distributed about EPMAR.

 

 

100

Program HARVSIM was used to calculate EPMAR values
(based on 80 growers) for harvest system 3 assuming a range
of initial equipment costs and harvest rates. The proba-
bility of negative PMAR was calcualted for each EPMAR
value and the results were plotted on probability paper, 1
Figure 8.1. The probability of negative PMAR was treated. E
as the independent variable, equipment cost as the l
dependent variable, and harvest rate as a graph parameter.

The results show the combinations of initial equipment

 

cost and harvest rate (other parameters held constant)
required for a given probability of negative PMAR.

In Section 8.1 it was concluded that a low proba-
bility of negative YMAR was also required. Probability--
parameter graphs can be prepared for each year of the
planning period using the results generated by HARVSIM.
However, the year of most interest will be the last year
of the planning period since it includes the most recent
historical data. The results for probability of negative
YMAR in 1971 are shown in Figure 8.2.

The results in Figures 8.1 and 8.2 apply when the
initial assumptions on other parameters (see Chapter 7)
are held constant. However, two other important design
parameters, machine recovery and fruit injury, can change.
To determine their affect on probability these parameters

were varied, in the above simulation, for harvest rates

101

 

 

 

 

     

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102

 

 

 

 

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103

of 10 and 15 trees per hour. The results, Figures 8.3

and 8.4, show the combination of initial equipment cost,
harvest rate, machine recovery, and fruit injury required
for a given probability of negative YMAR. Similar graphs
can be prepared for the probability of negative PMAR.
However, if parameter values are selected so that the
probability of negative YMAR is low (e.g., 15%), the
probability of negative PMAR will automatically be less
than the probability of negative YMAR. The following

facts provide proof for this statement: (1) PMAR is the
sum of YMAR; (2) a low probability of negative YMAR means
that EYMAR must be positive, thus EPMAR will be positive;
(3) the standard deviation of PMAR is only slightly larger
than the standard deviation of YMAR; (4) the ratio of EPMAR
to the standard deviation of PMAR will always be greater
than the ratio of EYMAR to the standard deviation of YMAR--
thus, the probability of negative PMAR will always be less
than the probability of negative YMAR.

Figures 8.3 and 8.4 again show that small changes
in machine recovery and fruit injury result in large
changes in YMAR. Also, the magnitude of change in proba-
bility or in initial equipment cost for a change in
machine recovery or fruit injury is less at a high
harvest rate than at a low harvest rate. Obviously this
process could be continued until variations in all con-
trollable or design parameters are included in similar

graphs.

.0 - ‘-

 

 

 

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104

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106

8.3 Procedure for Selecting Design
Parameter Values

 

All relations discussed in the previous section
were based on historical data. In equipment design the

conditions which will exist when the equipment is put into

 

use must be anticipated. Each year of the planning period V

will have different conditions, thus some guesswork is

required. E
The trend of annual results for both mechanical 1

harvest systems analyzed, Tables 7.3 and 7.5, indicate that L

a low probability of negative PMAR does not require a low
probability of negative YMAR each year. However, in
equipment design it seems reasonable to require a low
probability of negative YMAR for conditions considered
likely to occur. This suggests that parameter value
selection should be based on a low probability of negative
YMAR.

One approach would be to estimate the equipment
cost, labor cost, and fruit values which may exist when the
harvester is ready for use, then use HARVSIM to determine
simulated results for one year, such as given in Figure
8.4, and select the required parameter values. For example,
if the results in Figure 8.2 had been determined using
eStimates for 1972 and a 10% probability of negative YMAR
was used as the design criteria, a mechanical harvester
with a projected cost of $23,800.00 in 1972 would need

to have a harvest rate equal to or greater than 17 trees

107

per hour. Contrast this combination with a criteria of
50% probability of negative YMAR (breakeven) where the
same equipment cost requires a harvest rate of only 11
trees per hour.

A second, less accurate, approach would be to use F
the most recent probability and design parameter relations,
such as given in Figure 8.2, then estimate the‘required
values for the design parameters so that the probability

of negative YMAR is at some acceptable level (e.g., 10%).

 

"n...

Next, use the sensitivity analysis results to determine if
EYMAR will change substantially due to projected increases
in equipment cost and wage rates, or possible changes in
fruit values, and to select a final set of values for the
design parameters. A simple example will illustrate this
approach.

Suppose, using 1971 data4 a proposed harvest system
had the relations given in Figure 8.2. For a 10% proba—
bility of negative YMAR a mechanical harvester costing
$15,200.00 in 1971 would need to have a harvest rate equal
to or greater than 10 trees per hour. If wage rates and
equipment costs increase by 5% for 1972, EYMAR will change

by the following amounts:

108

EYMAR for 1971, results of simulation $1702.00

EYMAR change for 5% hourly wage
increase, Figure 7.11 -179.00

EYMAR change for 5% piece rate

increase, Figure 7.12 +570.00

EYMAR change for 5% equipment cost 0'
increase, Figure 7.1 -300.00
Estimate of EYMAR for 1972 $1793.00

Fruit values were low for 1970 and 1971, Table 7.1. A

If fruit value should increase by $1.00 per cwt for pro-

 

F‘.

cessing apples and $0.50 per cwt for juice apples EYMAR
would decrease by $600.00, Figure 7.10. Thus, the final
estimate of EYMAR for 1972 would be $1193.00. This
$509.00 decrease in EYMAR could be recovered by designing
for a harvest rate of about 11.2 trees per hour, Figure
7.3.

In the opinion of the author, a low probability of
negative YMAR (in the range of 10-20%) should be used as
the primary design criteria. However, both before and
after the design parameter values are selected HARVSIM
should be used to determine results for historical data,
such as those in Tables 7.3 and 7.5, so that the conse--
quences of using a proposed harvest system can be more
fully understood. The use of this criteria together with
good estimates of future conditions should result in a

very low probability of negative PMAR in actual practice.

109

8.4 Additional Relations

 

Graphs of the probability--parameter relations,
such as Figures 8.1-8.4, can provide additional informa-
tion. For example, since EYMAR and EPMAR are linearly
related to initial equipment cost, right-hand ordinates F
for these variables can be added to each figure so that 7
their values can be directly estimated. A right-hand g
ordinate giving EYMAR values for the 10 tree per hour line

(a different EYMAR scale must be used for each line) has

 

been added to Figure 8.3. If the initial assumptions
(Section 7.1) are met for harvest system 3, the probability
of negative YMAR is 12% and EYMAR is $1702.00. However,

if machine recovery should decrease .04 and fruit injury
increase .04 the probability of negative YMAR is 55%. The
correSponding EYMAR is -$135.00, but to avoid confusion

its scale is not shown.

Because YMAR is symmetrically (normally) distributed
about EYMAR, a 12% probability of negative YMAR for an
EYMAR of $1702.00 also implies a 12% probability that YMAR
will be greater than $3404.00. Also, by symmetry, the
probability of YMAR greater than $1702.00 or less than
$1702.00 are both 50%.

If a grower purchased harvest system 3 in 1971
the initial cost would be $18,200.00 (due to 5% inflation
per year). Figure 8.2 shows the probability of negative
YMAR for that grower would be 24% if all initial assumptions

were met. A harvest rate of 11.2 trees per hour would be

110

required to achieve a 10% probability of negative YMAR.
This indicates that if the same rate of inflation occurs
in equipment cost and wage rates, a delay in purchasing
a harvester will result in a higher level of required
performance if a given probability of negative YMAR is r
desired.

The EYMAR and probability of negative YMAR cited

 

for harvest system 3, at an initial equipment cost of

$15,700.00, are slightly different from those given in

 

Tables 7.3 and 7.5 for the same assumptions. The dif—
ference is due to the use of a different series of random
numbers (i.e., a different sample of 80 growers). By
considering 1 standard deviation of the mean of YMAR (160)
it is obvious that these two simulations are reasonable
samples from the same population. The difference in
probability of negative YMAR is about 7% and is due almost
entirely to the difference in the standard deviation of
YMAR for the two simulations. This difference in prob-
ability indicates that a design criteria of less than 10%
probability would not be reasonable.

8.5 Advantages, Disadvantages,
Limitations

 

 

The advantages of using simulation and the pro-
posed criteria for selecting design parameter values

instead of deterministic methods may be summarized as:

111

Measures of risk are provided in addition to
deterministic measures of benefit.
The requirement of a low probability of economic loss
(i.e., a high probability of economic gain) can be
used as a design criteria.
Use of this method should increase confidence that
a prOposed harvest system will be beneficial.
Research results should be applicable to a greater
number of intended users.

The disadvantages may be summarized as:
More input data is required.
More time is required for analysis.
A higher design cost is incurred.
Computer facilities are required for analysis.
Design requirements are set at a higher level.

The general limitations of the present method are:
Not all interactions, or "real world" correlations
are included in the present simulation model.
The method is new and untried. A few years of
experience will be necessary to determine if the above
advantages can be realized.
Care must be exercised in using simulation results
which have not been validated. However, the results
obtained from the simulation models designed are
thought to be realistic and to provide substantially

more guidance than do deterministic calculations.

 

9. SUMMARY AND CONCLUSIONS

Apple growers in Michigan are interested in

x’ " I ..TWI I

feasible mechanical harvest methods for processing apples.
If harvesting was mechanized, increased productivity would

permit hourly wages to be increased and would eliminate

 

the peak demand for harvest labor.

Many of the variables in apple production, harvest-
ing, and marketing are of a stochastic nature. Known
functional relations for the individual grower were used
to design a harvest simulation model, HARVSIM, to analyze
some of the benefits and consequences of proposed
mechanical harvest systems. Output statistics for each
proposed harvest system include the expected value and
standard deviation for margin, planning margin, harvest
cost per bushel, productivity, and indices for the volume
of fruit in processing, juice, and windfall quality
categories.

Sub-system models, required in HARVSIM, were
designed for annual yield, daily windloss, daily work time
lost due to rain or Sundays, and machine breakdown.

The model for annual yield generates yields via

a first-order autoregression relation which uses the

112

113

cummulative distribution function for annual yield and
negative correlation between successive yields for a given
variety.

The model for daily windloss generates average
daily wind velocities via a first-order autoregression
relation which uses the cummulative distribution function
for average daily velocity and positive correlation between
successive velocities. A derived relation between daily
windloss and daily average wind velocity is used to
calculate daily windloss for a given variety.

The model for daily work time lost due to rain
generates daily lost time using the cummulative distribu-
tion function for daily lost time based on historical
records of hourly rain observations and a no-work criteria
based on amount of rain. The first Sunday each season
occurs at random within the first seven days. Successive
Sundays occur at seven-day intervals.

The model for machine breakdown assumes that

operating time between failures is exponentially distributed.

Harvestrate--acreagerelationships'were determined
based on the length of harvest season, beginning and ending
dates for harvesting each variety, harvest rate, occurrence
of lost time, and probability of completing the total

acreage within the defined length of season.

Delayed harvest policies were evaluated to determine

if harvested volume would change substantially under the

combined affects of fruit growth and daily windloss.

 

 

114

For a planning period including years 1968-1971,
HARVSIM was used to simulate results for 80 growers with
70 acres of standard type apple trees and two possible
mechanical harvest systems. A sensitivity analysis,
using variations in 12 input parameters, was conducted for F“
both harvest systems. L

Three criteria for the selection of design

parameter values were examined: (1) breakeven conditions

 

 

between hand and mechanical harvest methods; (2) a low
probability of negative margin; and (3) a low probability
of negative planning margin.

Simulation and sensitivity results were used to
develOp a procedure for selecting values for harvester
design parameters so that a low probability of negative
margin can be achieved.

Conclusions derived from this study included:

1. The models develOped for annual yield and daily windloss
are adequate for harvest simulation applications.

2. The model develOped for daily work time lost due to
rain agreed closely with grower Opinion.

3. For a 90% probability of completing the harvesting of
70 acres of standard type trees in a six-week season,
a harvest rate of 10 trees per hour is required. This
is based on six working days per week and a maximum of

eight hours of work per day. The effect of work tune

115

lost due to Sundays, rain, and machine breakdown is in-
cluded. A change in total acreage requires a proportion-
ate change in harvest rate.

Simulation results for a general policy of delaying the

start of harvest indicates that the harvested volume

'J!‘.L" "I'D. I l.‘

will decrease slightly, except under conditions of sus-
tained rapid fruit growth.
Simulation results indicate that the expected margin for

two assumed mechanical harvest systems increased stead-

 

ily and the probability of negative margin decreased
steadily during the period 1968-1971.

Sensitivity analysis indicates that small variations
in harvest rate, machine recovery, fruit injury, and
hand picking cost cause large variations in margin
planning margin.

An important part of the simulation model is the in-
clusion of negative correlation between successive yields.
This feature adds realism, and when expected planning
margin is positive, the probability of negative plan-
ning margin is less than would result if successive
yields were independent.

A low probability of negative margin (in the range

of 10-20%) is prOposed as a criteria for selecting de-

sign parameter values for mechanical harvest systems.

116

Compared to the commonly used breakeven criteria,
this criteria will provide a higher level of abund-
ance that a proposed harvest system will be of econo-
mic benefit.

Relations between the probability of negative margin
and various combinations of design parameter values

can be easily described by graphical methods.

 

 

10 . RECOMMENDATIONS

 

HARVSIM and the proposed procedure for selecting

design parameter values should be regarded as a first

".2 '9"_I( 3 - I—F
=I
‘1'

attempt at applying simulation techniques to mechanical

Tr 1 ,

harvesting research planning, system evaluation, system

design, and system management. Refinements and differ-

 

ent approaches should undoubtedly be considered.
Additional "real world" inter-relationships should be
considered for inclusion in HARVSIM. For example,
relations between fruit price and annual yield, daily
windloss and number of days since reaching maturity,

fruit growth and percent of crOp with acceptable size

 

for processing, processor Operation and harvest method
(fruit price may be affected), may improve the realism
and usefulness of the model.

Detailed data on machine recovery and fruit injury
should be obtained from experimental harvesters to
determine if these parameters may be considered as
constants or must be considered as random variables.

A simulation model should be designed for analyzing
the mechanical harvesting of apples for fresh market
utilization. The model HARVSIM is a logical starting

point for a fresh market model.

117

118

Researchers working on the development of mechanical
harvest systems for fruit and vegetable crops should
make use of simulation and the harvest system design
procedure proposed in this study. The major reasons
for this recommendation are that by using these
techniques:

a. Harvest rate--acreage relationships can be
accurately determined.

b. Unknown relations, such as the windloss-wind
velocity relation, can be inferred from available
data.

c. Theoretical differences between various policies
can be determined.

d. Standard deviations for margin, and other useful
measures of goodness, can be determined.

e. Harvest systems can be designed and evaluated

using a criteria of low risk of economic loss.

 

 

 

REFERENCES

119

 

10.

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I 3' __N ’1

Batjer, L. P. 1954. Plant regulators to prevent
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