Description
Gathering the harvest is a complex managerial problem in controlling risk. Since the dawn of agriculture about 12,000 years ago, farmers have dealt with several categories of risk in producing the food needed to sustain their lives and to build urban centers of industry and learning.
MANUFACTURING & SERVICE
OPERATIONS MANAGEMENT
Vol. 6, No. 3, Summer 2004, pp. 225–236
issn1523-4614 eissn1526-5498 04 0603 0225
informs
®
doi 10.1287/msom.1040.0035
©2004 INFORMS
Controlling the Risk for an Agricultural Harvest
Stuart J. Allen
Penn State University, [email protected]
Edmund W. Schuster
MIT Auto-ID Labs, 23 Valencia Drive, Nashua, New Hampshire 03062, [email protected]
G
athering the harvest represents a complex managerial problem for agricultural cooperatives involved in har-
vesting and processing operations: balancing the risk of overinvestment with the risk of underproduction.
The rate to harvest crops and the corresponding capital investment are critical strategic decisions in situations
where poor weather conditions present a risk of crop loss. In this article, we discuss a case study of the Concord
grape harvest and develop a mathematical model to control harvest risk. The model involves differentiation of
a joint probability distribution that represents risks associated with the length of the harvest season and the
size of the crop. This approach is becoming popular as a means of dealing with complex problems involving
operational and supply chain risk. Signi?cant cost avoidance, in the millions of dollars, results from practical
implementation of the Harvest Model. Using real data, we found that the Harvest Model provides lower-cost
solutions in situations involving moderate variability in both the length of season and the crop size as compared
to solutions based on imposed risk policies determined by management.
Key words: harvest risk; agriculture
History: Received: July 30, 2002; accepted: December 12, 2003. This paper was with the authors 7
1
2
months for
6 revisions.
1. Introduction
Gathering the harvest is a complex managerial prob-
lem in controlling risk. Since the dawn of agriculture
about 12,000 years ago, farmers have dealt with sev-
eral categories of risk in producing the food needed
to sustain their lives and to build urban centers of
industry and learning (Garraty and Gay 1972, p. 50).
For an agricultural cooperative in modern times,
the harvest is the primary source of assets needed to
run the business. Hence, the harvest represents the
base raw material and most important link in the sup-
ply chain for these ?rms. Before each harvest, man-
agers must make critical assessments concerning the
capability of harvesting the entire crop at optimum
maturity.
In this article, we analyze the risk associated with
the harvesting of Concord grapes through the devel-
opment of a mathematical model and the presentation
of results from practice. Speci?cally, the model calcu-
lates the optimal rate for a processing plant to receive
grapes. This is a common situation in agriculture
where growers deliver crops to a central receiving
area for further processing and storage prior to sale.
Determining the optimum rate to harvest and process
the crop is a challenging problem in balancing the
risk of overinvestment in capital with the risk of not
harvesting the entire crop.
2. Structure of the Concord
Grape Industry
The Concord grape, a ?avorful, aromatic purple vari-
ety of grape, is grown in the cooler regions of the
United States. Major growing areas include western
New York, northern Ohio, and northern Pennsylva-
nia (all three near Lake Erie); western Michigan; and
south-central Washington state. In 1869 Thomas B.
Welch, a dentist from New Jersey, used pasteuriza-
tion as a means to preserve Concord grape juice. By
heating the grape juice before bottling he was able to
stop fermentation, creating the ?rst nonalcoholic fruit
juice product. This event marked the beginning of the
bottled and canned fruit juice industry in the United
States.
During the 130-plus years since this genesis, the
market for Concord grapes has grown into a billion-
dollar industry with an annual harvest of about
225
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
226 Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS
400,000 tons. Though the Concord grape comprises
only about 6% of the total grapes grown in the United
States, it maintains a high pro?le because of intensive
marketing; it remains a popular ingredient for juices,
fruit jellies and jams, and concentrates. Not suitable
for the fresh market because of its short shelf life,
nearly all of the Concord grapes harvested are imme-
diately converted into juice.
The leading player of the industry is Welch Foods,
Inc., which is owned by the National Grape Cooper-
ative Association. Welch Foods controls about 55% of
the Concord grapes grown in the United States. The
next-largest player controls less than 7%. Collectively,
?ve business structures comprise the Concord grape
industry.
(1) Private packers that purchase Concord grape
juice as a major ingredient for their products, both
branded and house brands.
(2) Private processors that contract with growers on
an annual basis for delivery of grapes, then process the
grapes into concentrate for sale as a raw ingredient.
(3) Private processors with packaging capabilities, pri-
marily involved in production of house brands.
(4) Cooperatives owned by growers that process
grapes into raw ingredients, sell to other companies,
and return pro?ts to the growers.
(5) Vertically integrated cooperatives involved in
growing, processing, packaging, distributing, and
marketing Concord grape products.
The industry is a mix of both vertically integrated
and modular structures (Fine 1998). However, the
long-term trend is toward supply chain integration
similar to the transition taking place in the poultry
industry (Kinsey 2001). Product life cycles are long,
with few radical product innovations. Intense compe-
tition and lack of pricing power in the market com-
bine to force a focus on cost control.
2.1. Operational Design, Organization, and
Financial Implications
All businesses that convert Concord grapes into
juice—cooperatives and private companies alike—
follow similar steps during the harvest process. In the
fall, usually beginning in late September, fruit grow-
ers deliver grapes picked by mechanical harvesters
to processing plants located near the vineyards. The
grapes are then converted into juice through a com-
plex pressing process. This step takes place within
eight hours after picking to avoid fermentation and
deterioration of quality. Exceeding this time limit
means the grapes must be diverted, at severe cost
penalty, to other uses, such as wine making. The
economics of refrigerated warehousing, as well as
biological degradation at low temperatures, preclude
storage of raw Concord grapes for later pressing into
juice.
In each growing area, approximately 100 mechan-
ical harvesters pick grapes on a continuous basis.
Since the cost of an individual harvester is very
high, approaching $200,000 per unit, growers usually
pool resources through joint ownership or contract-
ing. A single harvester picks an average of three sep-
arately owned vineyards. Coordination of the ?eld
operations becomes critical to ensure a steady feed of
freshly picked grapes to the processing plant.
2.2. Harvest Scheduling and Capacity
Prior to the harvest, planners assign a constant rate of
picking to each harvester operator based on acreage,
estimated yields, the estimated length of the har-
vest, and the standard pressing rate for the receiving
plant. This practice establishes fairness: Each vineyard
owner faces the same risk of losing crop because of
poor weather, such as a frost. A hard frost, where tem-
peratures dip below 28
F, weakens stem tissue, caus-
ing the grapes to fall from the vines. This is called
shelling. Every grower wants to complete the harvest
before a hard frost causes shelling in the vineyards.
The total daily schedule for all harvesters equals
80% of the standard plant pressing rate. The reserve
of 20% allows planners ?exibility during harvest
operations to accommodate individual growers who
might experience crop maturity problems, mechani-
cal breakdowns, or an abnormally high risk of crop
loss due to frost. The reserve is completely used as
part of day-to-day harvest operations so that the rate
of picking always matches the rate of pressing at the
receiving plant. This implies that the rate of picking
and pressing is dictated at the same time that capac-
ity investment decisions are made (i.e., once per year,
projecting ?ve years into the future).
In the midst of harvest operations, it is very dif-
?cult to change the rates of picking and pressing
grapes because of the large number of harvester oper-
ators and the complexity of receiving plant opera-
tions. When day-to-day pressing or picking rates vary
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS 227
by even small amounts a classic system dynamics
response occurs, producing substantial loss of ef?-
ciency that extends many days past the original dis-
ruption. Extensive preharvest efforts focus on testing
and maintenance of plant equipment to ensure relia-
bility in achieving the standard pressing rate per day.
Upon receipt from the vineyards, the grapes are
heated and pressed into unsettled juice that is stored
in large refrigerated tanks. Unsettled juice is the
intermediate step in processing. Welch Foods alone
has over 55 million gallons of tank storage capacity.
Throughout the remainder of the year, the unsettled
juice is further processed and used for bottling and as
an ingredient in other food products. The manufac-
turing structure takes on a “V” shape typical in the
process industries where a small number of raw mate-
rials are used to produce a large number of end items
(Umble 1992). Final processing of the entire harvest
takes close to one year.
2.3. Financial Implications Investment in
Capital Assets
Investment in capital assets to process Concord
grapes into juice is substantial. For cooperatives this
investment in pressing capacity, that remains idle for
10 months per year, is a drain on resources. Despite
tax advantages, cooperatives involved in processing
raw fruits and vegetables typically have a low return
on capital compared with the rest of the food industry.
In this case, the proper sizing of harvest processing
capacity becomes an important strategic decision with
signi?cant ?nancial impacts.
3. Problem De?nition
Consider the typical situation in harvesting Concord
grapes, or other crops, where a planner faces a task
with an uncertain start date, an uncertain end date,
and whose size cannot be known in advance with
certainty. As an example, the maturation dates of
Concord grapes depend on random factors such as
weather and soil conditions as well as cultivation
practices. Random frost dates may signal an abrupt
end to the harvest season and total crop size is not
known in advance of the harvest process.
In such cases, the planner must decide what degree
of effort to devote to the harvest in terms of the press-
ing rate and the corresponding capital investment in
equipment. These are decisions made in advance of
harvest as part of rolling ?ve-year plans because of
the long lead times for capital equipment. A balance
must be made between the costs of the resources com-
mitted and the costs of a partially completed task
(grapes remaining in the vineyard). With whatever
methods employed, the derivation of the optimal har-
vest rate is critical for the grower in controlling risk
and in determining the proper capital investment.
3.1. The Elements of Uncertainty
With uncertainty in both the size of future grape crops
and the time window available for future harvests, it
is impossible to calculate the optimal rate of pressing
using direct methods. Simply dividing the historical
average crop size by an estimate of the available days
of harvest will result in a solution that ignores the
inherent risk of agriculture.
The essence of this problem can be de?ned by three
variables:
H = crop size in tons, a random variable;
| = length of harvest season in days, a random
variable;
í = pressing rate in tons per day, the decision variable.
From these variables two sets of expressions can be
de?ned, one set each for the amount of crop loss (ACL)
and the excess harvest rate (EHR).
ACL =
H ?RL if H >RL
0 if H ?RL
(1)
EHR=
0 if H ?RL
RL?H if H -RL.
(2)
These two sets of expressions have natural interpre-
tations. If H is greater than what can be harvested at
í, then there will be loss in the vineyards, as repre-
sented in Equation (1). This often occurs in practice
when a frost cuts short the harvest season. How-
ever, if í is too high, the opportunity is lost for har-
vesting the crop with less effort, as represented in
Equation (2). This also occurs in practice when there is
overinvestment in pressing equipment. Because both
of the above equations contain two random variables,
we will have to form the expected values of each by
integrating over the joint density function, which we
denote by g(H, |).
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
228 Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS
3.2. Important Model Assumptions
and Justi?cations
Allen and Schuster (2000, pp. 32–33) report that for
one of the Concord grape growing regions a correla-
tion coef?cient of 0.14 exists between H and | based
on 22 years of data at an observed signi?cance of
53%. The results of this analysis offer support that H
and | are statistically independent. This makes sense
because H is dependent on many variables such as
precipitation, soil conditions, temperatures, and avail-
able sunlight, which are uncorrelated to the time win-
dow of good weather available to gather the harvest.
Then the joint density function can be formed as the
product of two marginal distributions.
g(H, |) =] (H) ×] (|). (3)
Another important assumption is the shape of
the probability distributions used to estimate H and
|. Extensive historical information from state agri-
cultural experiment stations, spanning more than
40 years of history, allows calculation of the mean
and standard deviation in each case. Histograms
of this data show that both H and | are mound
shaped and reasonably symmetrical. An analysis of
data from Allen and Schuster (2000, pp. 32–33), using
a chi-square goodness-of-?t test, shows insuf?cient
evidence (p = 0.53 and p = 0.66) to reject the null
hypothesis that H and | are normally distributed. In
both cases, one degree of freedom exists. The modi-
?ed histograms of H and | contain four observations
in one and two intervals, respectively. This is out-
side the minimum guideline of ?ve observations for
the chi-square goodness-of-?t test. It is also impor-
tant to note that in similar situations researchers use
symmetrical distributions (Jones et al. 2001). These
observations lead us to assume | and H are normally
distributed with mean and standard deviation j
|
, u
|
and j
H
, u
H
respectively.
3.3. The Objective Function
The Harvest Model minimizes a total cost function
that contains two costs.
• Underage cost (C
H
, dollars per ton). This re?ects the
cost of harvesting too slowly, therefore foregoing sales
of grapes left in the ?eld at the end of the harvest
season.
• Overage cost (C
í
, dollars per ton). This re?ects the
cost of harvesting too quickly, therefore committing
an excessive amount of resources to the harvesting
effort.
The formation of the Harvest Model involves
applying costs to ACL and EHR. Because the Har-
vest Model is probabilistic in nature, expected values
are needed for both ACL and EHR. Once expressions
exist for each, costs can be applied to form a total cost
function. The minimum of this function will provide
the optimal harvest rate at minimum cost.
The remainder of this study will focus on the mat-
ter of harvesting Concord grapes with an uncertain
length of harvest season and an uncertain crop size.
Section 4 provides an overview of the literature relat-
ing to the Harvest Model. Section 5 gives speci?cs
about the derivation of the expected values and the
minimization of the total cost function. In §6 detailed
numerical results of the Harvest Model are presented
based on real data, followed by §7, which provides
managerial insights.
4. A Review of Cognate Research
The analysis of harvest risk draws upon three gen-
eral areas of previous research, including the analysis
of harvesting operations, agricultural decision making
involving risk, and the newsvendor model.
4.1. Analysis of Harvest Operations
Thornthwaite (1953) addresses the problem of har-
vesting peas at the peak of maturity. Subsequent
analysis by Kreiner (1994) shows that Thornthwaite’s
method was still in use more than 40 years later
and has reduced harvest and processing costs, and
lost product. Though Thornthwaite’s work involves
annual crops (peas), and does not deal with the prob-
ability of losing crop as a result of frost, it remains
a classic in the application of operations research. To
our knowledge this is the ?rst reference on the scien-
ti?c study of harvest operations.
Porteus (1993a, 1993b) develops a two-part case
study of the National Cranberry Cooperative. In this
case, he examines complex trade-offs in capital invest-
ment and capacity for the processing of cranber-
ries during a harvest season. Although the harvest
and processing of cranberries share common elements
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS 229
with Concord grapes, the author did not address the
risk of crop loss from a frost, or from overmaturation.
In aggregate, these four references provide impor-
tant background information on harvesting opera-
tions plus speci?cs about the uniqueness of this area
of management.
4.2. Agricultural Decision Making
Involving Risk
Maatman et al. (2002) note that “various methods
of risk reduction exist, including those aimed at pre-
vention of risk (e.g., irrigation), dispersion of risk (by
diversi?cation of risky activities such as the cultiva-
tion of different varieties of crops), control of risk (e.g.,
by sequential decision making) and ‘insurance’ against
risk” (p. 400, italics in original).
Much of research in agricultural decision making
focuses on the dispersion of risk through improved
individual farm planning. Mathematical program-
ming is often the tool used for modeling farmers’
strategies under uncertain conditions (see Anderson
et al. 1977, Hazell and Norton 1986). These models are
static in nature, assuming that probabilities of occur-
rence are known in advance and that all decisions are
made at one time.
This class of models is similar in spirit to what is
presented in this article. However, the Harvest Model
does not focus on the dispersion of economic risk and
does not employ traditional mathematical program-
ming. Rather, we concentrate on control of risk for a
pooled harvest situation (a cooperative) through opti-
mization based on a generalization of the newsvendor
problem.
Jones et al. (2001) study the control of risk for pro-
duction of hybrid seed corn from the perspective of
both random yields and demands. A follow-up study
reports results in practice (Jones et al. 2003). The
authors’ work is similar in style to the Harvest Model.
However, the authors do not address uncertain start
and end times for a harvest, choosing instead to focus
on a two-stage production model that involves two
different regions (North America and South America)
with staggered growing seasons. In this regard, their
work differs from the Harvest Model, and will no
doubt lead to further work in controlling agricultural
risk.
Previous research involving Concord grapes and
agricultural cooperatives deals with decision making
in raw materials management (Schuster and Allen
1998, Schuster et al. 2000) and production planning
(Schuster and Finch 1990, Allen and Schuster 1994,
Allen et al. 1997, D’Itri et al. 1998). Though these mod-
els yield signi?cant ?nancial savings in practice, none
considers the risk associated with harvesting Concord
grapes.
In a recent article, Allen and Schuster (2000) add-
ress harvest risk for Concord grapes through a work-
ing model with the successful outcome of a closed
form solution. Using this approach, senior manage-
ment speci?es risk levels as policy. An example would
be “Harvest 100% of the crop, 85% of the time.”
The policy level becomes an important input. Despite
offering a large improvement over previous methods
that involved heuristics, this model does not include
an explicit treatment of the cost of lost crop or sea-
sonal operating costs.
4.3. The Newsvendor Model
The newsvendor approach is effective in a num-
ber of applications, including the defense industry
(Masters 1987), and is the underpinning of our har-
vest model. Nahmias (1997, pp. 272–280) provides
a comprehensive review of the newsvendor model
including derivation, historical context, and several
important extensions.
The process of harvesting grapes shares character-
istics with the management of style goods such as
fashion items in the clothing industry. As is true of
style goods, grapes are perishable. Concord grapes
have a limited time for peak maturity combined with
a short time window for harvesting during the fall. In
this regard, the harvest of Concord grapes shares sim-
ilarities to seasonal sales of clothing where there are
substantial costs of obsolescence and limited oppor-
tunities to store for future sale.
An important body of literature addresses the style
goods problem. Fisher and Raman (1996) develop a
response-based production strategy to deal with the
uncertain demand characteristic of style goods. Other
authors improve style goods forecasts by using a
Bayesian approach (Eppen and Iyer 1997, Murry and
Silver 1966). However, with the harvest of Concord
grapes, little opportunity exists to change the harvest
rate during seasonal operations, rendering Bayesian
methods less effective.
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
230 Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS
As a concluding comment, the distinguishing
aspect of the Harvest Model is that it contains two
random variables (H and |) as compared to the tradi-
tional newsvendor model that contains only a single
random variable (demand). In this way, the harvest
risk model shares similarities with the work of Song
et al. (2000) that considers two random variables.
5. Mathematical Development
We now turn our attention to expressions for the
expected ACL and EHR.
5.1. The Expected ACL
Since the ACL is discontinuous at H =RL, the expec-
ted value is given by
E(ACL) =
|=
|=?
H =
H =RL
(H ?RL)] (H) dH
] (|) d|.
(4)
The inner integral is the familiar normal loss function.
We de?ne the standard normal variate for H as
z
H
=
H ?j
H
u
H
, (5)
and
Q=
RL?j
H
u
H
. (6)
Then the expected ACL becomes
E(ACL) =u
H
?
G(Q)] (|) d| (7)
where
G(Q) =4(Q) ?Q|1 ?+(Q)] (8)
and 4(·) is the standard normal density function
while +(·) is the cumulative of the standard normal
density function. The form of the expected ACL can
be simpli?ed by introducing the nondimensional har-
vest rate, r. In a risk-free environment, the risk-free
harvest rate becomes í
0
= j
H
,j
|
. In this situation,
í is known because both H and | are known in
advance. Of course in reality uncertain crop size and
uncertain harvest start and end times exist, so í is
different from í
0
. The de?nition for the nondimen-
sional harvest rate becomes r =í,í
0
. Next, we de?ne
the standard normal variate for | as
z
|
=
|?j
|
u
|
. (9)
Finally, we de?ne the two coef?cients of variation
|
H
=u
H
,j
H
, |
|
=u
|
,j
|
.
Now Q can be expressed entirely in terms of the two
coef?cients of variation, the standard variate, z
|
, and
the dimensionless harvest rate, r:
Q=
r(1 +|
|
z
|
) ?1
|
H
. (10)
Then the expected ACL becomes
E(ACL) =j
H
|
H
?
G(Q)4(z
|
) dz
|
. (11)
Silver and Smith (1981) provide the tools to carry
out the integration of this expression and subsequent
results. First note that Q in Equation (10) can be
expressed in the form Q=uz
|
+|, where
u =r(|
|
,|
H
), | =(r ?1),|
H
. (12)
In the interest of notational conciseness, we de?ne
another recurring parameter as
c =|,
1 +u
2
. (13)
Using Equations (12) and (13), the integrated form of
Equation (11) becomes
E(ACL) =j
H
|
H
1 +u
2
G(c). (14)
In Equation (14) and all subsequent results, G(·) is
de?ned by Equation (8) and we emphasize that 4(·) is
the standard normal density function (i.e., zero mean
and unit variance) and +(·) is the cumulative of the
standard normal density function.
5.2. Expected EHR
Similar to ACL, this function is also discontinuous at
H =RL so
E(EHR) =
|=
|=?
H =RL
H =?
(RL?H)] (H)] (|) dH d|.
Now add and subtract the complement to the inner
integral and rearrange to get
E(EHR) =
|=
|=?
u
H
G(Q) +
H =
H =?
(RL?H)] (H) dH
· ] (|) d|,
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS 231
where Q is as previously de?ned in Equation (10).
The inner integral simpli?es to |RL?j
H
] and on sub-
sequent integration over | this becomes |íj
|
?j
H
].
This should be in nondimensional form as well, so
using earlier de?nitions, the result is
E(EHR) =j
H
|
H
?
G(Q)4(z
|
) dz
|
+j
H
(r ?1). (15)
The integrated form of Equation (15) becomes
E(EHR) =j
H
|
H
1 +u
2
G(c) +j
H
(r ?1). (16)
5.3. Expected Total Cost and Optimal Harvest Rate
The integral form of the total cost (TCP) per ton of
harvest can now be expressed as an expected value
using Equations (11) and (15):
E(TCF),j
H
= C
H
|
H
?
G(Q)4(z
|
) dz
|
+C
í
|
H
?
G(Q)4(z
|
) dz
|
+C
í
(r ?1). (17)
The integrated form of Equation (17) is
E(TCF),j
H
= (C
í
+C
H
)|
H
1 +u
2
G(c)
+C
í
(r ?1). (18)
Next, an expression is obtained for the optimal
dimensionless harvest rate. Taking the ?rst derivative
of Equation (17) with respect to r, setting the result to
zero, and rearranging terms yields an implicit expres-
sion for the optimal dimensionless harvest rate (r
?
):
?
|1 ?+(Q)](1 +|
|
z
|
)4(z
|
) dz
|
=C
í
,(C
í
+C
H
). (19)
Again using Silver’s results, the integrated form of
Equation (19) yields
+(c) +|
|
u,
1 +u
2
4(c) =1 ?C
í
,(C
í
+C
H
), (20)
where u and c are functions of the decision variable r
and the parameters |
H
and |
|
through Equation (12).
It deserves emphasis that r
?
depends only on the
nondimensional ratios |
H
, |
|
; and the cost ratio C
í
,
(C
í
+ C
H
). Furthermore, the second derivative of
Equation (17) with respect to r is positive for all pos-
sible z values so that Equation (20) will yield the min-
imum r.
5.4. Expected Percentage Crop Recovery
Beyond r
?
, the Harvest Model also permits calcula-
tion of the expected amount of crop recovery. As men-
tionedpreviously, the introductionof riskintoplanning
means that the harvest will be less than 100%.
We de?ne the expected percentage crop recovery by
E(PCR) = |expected amount of crop harvested
/mean harvest size] ×100.
The expected proportion of ACL is E(ACL)/j
H
. Then
E(PCR) =|1 ?E(ACL),j
H
] ×100. (21)
5.5. An Alternative Approach to Harvest Risk
Management
Some decision makers prefer a policy approach (as
opposed to a cost-based method) for dealing with risk
management. In an earlier study, Allen and Schuster
(2000) examine a policy approach. The objective here
is to explore the connections between that study and
the cost-based analysis in the present study. In this
alternative, instead of specifying cost penalties man-
agement speci?es a policy for the probability of har-
vesting the entire crop. It is important to emphasize
that this is not at all the same as specifying the pro-
portion of crop harvested as was addressed in the
previous section.
The probability of harvesting the entire crop is
found from
Prob(complete crop harvest)
?P(CCH) =P(H ?RL) =1 ?P(H >RL)
P(CCH) =1 ?
|=
|=?
H =
H =RL
] (H) dH
] (|) d|.
The inner integral is simply |1?í (RL)], so again using
the de?nition of Q in Equation (10) employed in our
earlier work, we obtain
P(CCH) =1 ?
?
|1 ?+(Q)]4(z
|
) dz
|
.
We will illustrate the use of this policy-based risk
management method for the case of a speci?ed
P(CCH) =0.85 (85%) in §6.2. Integration of the above
yields the following implicit equation for the dimen-
sionless harvest rate, r:
+(c) =P(CCH) =0.85. (22)
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
232 Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS
6. Numerical Results
This section draws from real data on the harvest of
Concord grapes documented by Allen and Schuster
(2000) as a basis for examples and analysis. Their
work deals speci?cally with the decision-making pro-
cess involving the harvest of Concord grapes for
Welch Foods. The authors were directly involved
in planning and managing harvest operations for a
period of 10 years.
6.1. Data
The model requires two sets of historical data: H
(tons) and | (days). Both H and | vary from year
to year. In each growing area, the state agricultural
experiment station keeps detailed records by year on
yields and crop size. Historical records also show the
harvest start date (when the crop is at the correct level
of maturity) and the ?rst 28
F day (hard frost) that
signi?es the end of the harvest. The value of | for
each harvest can be calculated for a growing region
by counting the days between the start and end dates.
Similar to the crop size, | varies from year to year.
Besides H and |, the Harvest Model requires two
costs, C
H
and C
í
. The calculation of C
H
is straightfor-
ward; if a grower does not harvest a ton of grapes,
then the relative value of the loss equals the open
market price. This is a conservative approach to val-
uation. The examples presented assume C
H
= $250
per ton of grapes. This is a long-term estimate of the
open market price. The market price of grapes will
change a considerable amount on a year-to-year basis
depending on industry crop size, carryover from the
previous year, consumer demand, and the pricing of
alternative raw ingredients.
The calculation of C
í
is a more complex matter.
Harvesting at an excessive rate means that the entire
harvest will take place well in advance of a hard frost.
If this happens, then í is too high and the long-term
investment in capital is excessive. In this situation C
í
relates to the amortized cost of the unneeded capacity.
It is unique to each type of business organization.
Estimating C
í
is dif?cult because it involves capital
equipment that is added in steps. There is no question
that the cost is nonlinear over large ranges of í. How-
ever, for small changes of í the cost becomes linear.
An estimate can be calculated on a cost-per-ton basis
by analyzing small increases of í, e.g., an increase of
10% or less and the corresponding capital required to
support the higher pressing rate. This was the proce-
dure used at Welch Foods.
For example, at one grape processing plant located
in Pennsylvania, increasing í by 180 tons per day
requires a capital investment of $1.5 million. During
the course of an average harvest, this increased rate
means that 5,400 extra tons of grapes can be pressed
before a hard frost. Assuming a 10 year life span,
the estimated amortized cost of this extra capacity is
$150,000 per year. If part or all of this extra capacity is
not needed, then C
í
becomes the cost of the incremen-
tal capacity divided by the additional tons harvested
at the higher rate ($150,000 per 5,400 tons), or about
$30 per ton of grapes. This represents the estimated cost
of incremental capacity occurring around the point in
time when a hard frost might occur.
6.2. Results
We now provide numerical results from the Harvest
Model for the following practical problems:
• calculation of r
?
under different scenarios,
• determination of the percent recovery for a spe-
ci?c r
?
, and
• ?nding r under conditions of an imposed harvest
policy.
Using the historical data for H and | along with
C
H
and C
í
as inputs, the Harvest Model calculates
r
?
providing the opportunity to analyze a number of
harvest risk scenarios.
Table 1 gives results for r
?
of various combinations
of |
H
and |
|
for a cost ratio, C
í
,(C
í
+ C
H
), of 0.10
using Equation (20). These were obtained by using the
Goal Seek utility in Excel with a tolerance of 0.0001 for
all calculations of r. The data show that r
?
increases
with increases in variability for H and |. This is a con-
sistent result. The greater the risk, the greater the r
?
.
Table 1 Optimal r for C
R
/C
R
+C
H
=01
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 13350 13564 13828 14195 14620
030 13880 14019 14291 14618 15004
035 14306 14449 14662 14954 15305
040 14586 14732 14879 15167 15489
045 14713 14860 15008 15239 15532
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS 233
Figure 1 Cost Ratio vs. r
?
for k
H
=025
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Optimal Dimensionless Harvest Rate, r*
C
o
s
t
R
a
t
i
o
C
R
/
(
C
R
+
C
H
)
k
L
=0.05
k
L
=0.15
k
L
=0.25
k
L
=0.35
k
L
=0.45
Additional descriptive results can be observed by
evaluating the left-hand side of Equation (20) over
ranges of r
?
for representative values of |
H
and |
|
.
The resulting values correspond to the cost ratios at
optimality. The set of graphs in Figure 1 are for a con-
stant |
H
of 0.25 and a range of |
|
values. An estimate
of r
?
can be made directly from the graph without
invoking a search engine by specifying the cost ratio
and the coef?cient of variation for the length of har-
vest season. This provides planners with a visualiza-
tion of the relationship between the cost ratio and r
?
.
Computations of expected percentage of crop
recovery using Equations (14) and (21) for the r
?
of
Table 1 (for C
í
,(C
í
+C
H
) = 0.10), appear in Table 2
below.
Increasing variability in crop size and length of season
means lower crop recovery. This re?ects the trade-off
Table 2 E(PCR) in %
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 9716 9708 9695 9679 9661
030 9596 9591 9580 9565 9548
035 9452 9446 9435 9421 9404
040 9275 9271 9256 9244 9227
045 9067 9063 9049 9035 9018
Table 3 Total Cost per Harvest Ton at Optimal r
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 1731 1804 1919 2066 2237
030 2204 2268 2369 2501 2659
035 2728 2785 2876 2997 3144
040 3300 3352 3436 3548 3686
045 3913 3962 4041 4147 4278
between the cost of unharvested crop and the cost
of an EHR. Table 2 is an important tool to explain
why 100% recovery of a crop is not optimal given
the trade-off between crop loss and a consistent early
completion of harvest operations.
The total cost per ton for each value of r
?
can be
calculated using Equation (18). These results appear
in Table 3. As is the case with r
?
, the data show that
cost per ton increases with increases in variability for
H and |.
Finally, Goal Seek can be used to ?nd r for speci?ed
values of |
H
and |
|
under an imposed policy for crop
recovery, Equation (22). Decision makers often use
this approach to demonstrate how r changes for pol-
icy alternatives (suboptimal values of r) approaching
a complete harvest. Since a policy alternative might
not be optimal, it is important to understand the cost
difference between the optimal solution and the cho-
sen policy. In some cases, choosing an r that is not
optimal is justi?ed based on the timing of capacity
additions in relation to legacy equipment, an estab-
lished consensus by growers who are risk adverse,
and the uncertainty surrounding cost data or a shift in
historical trends. By substituting the values of r into
Equation (18) total costs can be computed for each
imposed policy alternative. These results appear in
Tables 4 and 5. The imposed policy for this example
is “Harvest 100% of the crop 85% of the time.”
Table 4 Policy r for 85%
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 13547 13682 13927 14225 14573
030 14570 14684 14883 15144 15458
035 15729 15842 16001 16249 16533
040 17094 17265 17371 17545 17837
045 18716 18903 19002 19192 19384
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
234 Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS
Table 5 Total Cost in US$ per Harvest Ton for Policy r
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 1733 1805 1919 2066 2238
030 2226 2286 2384 2513 2667
035 2798 2852 2936 3051 3191
040 3472 3528 3598 3696 3827
045 4275 4332 4394 4487 4596
Table 6 shows the cost penalties for the 85% policy
as compared to r
?
for each case of |
|
and |
H
. The cost
penalty for the 85% policy can be computed using the
following formula:
percent cost penalty = (policy cost–optimal cost)
· 100,optimal cost.
At the extreme (|
|
= 0.45), the maximum cost
penalty for the 85% policy is about 9.2%. For lower
values of |
|
, the penalty is less than 3%. It is inter-
esting to note that in all cases for a given |
|
the cost
penalty decreases as |
H
increases. The greater vari-
ability in crop size has the impact of narrowing the
cost difference between r, at 85% policy, and r
?
. Since
in all cases the penalties are modest, we conclude that
the 85% policy gives an r that is close to optimum in
terms of cost (for the set of parameters examined).
The Harvest Model also provides results based on
various policy alternatives that are close to those
obtained in the study by Allen and Schuster (2000)
with some small differences in the second decimal
place for the lower values of |
H
. The more accurate
values are probably those of the prior study that did
not rely on Goal Seek. This validates the Harvest
Model. The above results are not optimum from the
point of view of minimizing costs but rather are the
result of an imposed policy. This is an effective way
of communicating harvest risk to growers investing
in receiving plants through agricultural cooperatives.
Table 6 Cost Penalty for Policy r in %
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 014 007 003 000 000
030 096 081 063 046 031
035 256 239 207 181 150
040 522 525 472 417 382
045 923 934 874 819 743
The Harvest Model gives consistent answers to a
number of important questions involving risk. Based
upon considerable use in practice, the Harvest Model
is suitable as a general solution for problems encoun-
tered in agriculture where a sudden event, such as a
fall frost, concludes the harvest season.
7. Managerial Insights
The true test of any model is performance in practice.
The Harvest Model consistently provides reasonable
results in addition to new insights. As an example,
the traditional belief that the Concord harvest sea-
son should last 30 days to ensure complete harvest
of the crop proves inaccurate when analyzed using
the Harvest Model. Regional climatic variation plays
a signi?cant role in the probability of a frost. A large
body of water, such as Lake Erie, acts as a heat sink in
the fall, thereby extending the probable length of the
harvest season for growers in close proximity. Anal-
ysis using the Harvest Model shows that extending
seasonal operations beyond 30 days in parts of the
Lake Erie region can be accomplished with low risk
of crop loss. This translates into a slower harvest rate
and reduced capital investment for cooperatives in
this growing area.
However, similar analysis for growers in Michigan,
who are located a considerable distance from any
large body of water, shows that seasonal operations
should be less than 30 days because of signi?cant frost
risk. In this growing region a small probability exists
that a frost will occur before the beginning of the har-
vest. Given this case, the harvest rate must increase,
which requires more capital investment.
The authors have experience implementing results
of the Harvest Model, using normal probability dis-
tributions to express harvest size and length of sea-
son, for growing areas near Lakes Erie and Michigan.
In total, capital avoidance equaled $2 million through
establishing the proper harvest rate by region. This
is one example of the bene?ts of controlling harvest
risk, and of the adoption of precision agriculture in
practice.
Beyond savings in capital, the Harvest Model pro-
vides other organizational bene?ts. In the case of
Welch Foods, the Harvest Model is an important
tool for corporate planning with results of the model
being presented at meetings of the board of directors.
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS 235
Welch Foods is owned by the National Grape Coop-
erative and is an example of vertical integration in
agriculture. The company’s span of control ranges
from raw grapes to end-product marketing. A critical
measure of effectiveness is the capability of receiving
the grower’s grapes with little risk of loss. By using
the Harvest Model, management can communicate to
growers—the owners of the cooperative—exact levels
of risk associated with harvesting the crop. Shifting
dialogue to a fact-based discussion offers great advan-
tages in reaching a consensus concerning the proper
investment in capital for pressing operations.
In addition, the Harvest Model provides a struc-
ture to evaluate the impact on picking and pressing
operations from additions of vineyards to a coopera-
tive. As product lines expand and sales grow, the need
for Concord grapes also increases, causing additional
burden in harvesting. The decision to add incremental
acreage to the cooperative is in?uenced by the capa-
bility to harvest the crop at acceptable levels of risk
and capital investment. The Harvest Model provides
an effective structure to evaluate the change in risk
when additional vineyards are planted. Since risk lev-
els vary by growing region, the proper decision can
be made concerning the correct area to plant addi-
tional grapes. This is a safeguard against expansion
in high-risk growing regions where the probability of
a killer frost is high. Over the long run, this type of
prudent, risk-based decision making will result in an
increased level of reliability for raw material supply.
8. Conclusion
We live in a world ?lled with quanti?able risk, yet
there are few mathematical frameworks available to
guide practitioners who must deal with agricultural
harvest risk. Based on an extensive literature search,
we conclude that harvest risk is an undeveloped
area within operations management. Controlling risk
by the application of rational models presents many
opportunities to increase pro?ts.
In this article, we develop a robust mathematical
model to optimize the risk of an agricultural harvest.
We test the model with representative data from the
harvest of Concord grapes and ?nd it gives accept-
able solutions. During the course of our work we
also ?nd the model applies to other risk-bearing tasks
in agriculture, such as the harvest of oranges, and
in industries such as project management and style
goods planning. We plan further research in this area.
Acknowledgments
The authors wish to thank the editor, senior editor, and two
anonymous reviewers for the constructive comments pro-
vided during the review process.
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doc_899314630.pdf
Gathering the harvest is a complex managerial problem in controlling risk. Since the dawn of agriculture about 12,000 years ago, farmers have dealt with several categories of risk in producing the food needed to sustain their lives and to build urban centers of industry and learning.
MANUFACTURING & SERVICE
OPERATIONS MANAGEMENT
Vol. 6, No. 3, Summer 2004, pp. 225–236
issn1523-4614 eissn1526-5498 04 0603 0225
informs
®
doi 10.1287/msom.1040.0035
©2004 INFORMS
Controlling the Risk for an Agricultural Harvest
Stuart J. Allen
Penn State University, [email protected]
Edmund W. Schuster
MIT Auto-ID Labs, 23 Valencia Drive, Nashua, New Hampshire 03062, [email protected]
G
athering the harvest represents a complex managerial problem for agricultural cooperatives involved in har-
vesting and processing operations: balancing the risk of overinvestment with the risk of underproduction.
The rate to harvest crops and the corresponding capital investment are critical strategic decisions in situations
where poor weather conditions present a risk of crop loss. In this article, we discuss a case study of the Concord
grape harvest and develop a mathematical model to control harvest risk. The model involves differentiation of
a joint probability distribution that represents risks associated with the length of the harvest season and the
size of the crop. This approach is becoming popular as a means of dealing with complex problems involving
operational and supply chain risk. Signi?cant cost avoidance, in the millions of dollars, results from practical
implementation of the Harvest Model. Using real data, we found that the Harvest Model provides lower-cost
solutions in situations involving moderate variability in both the length of season and the crop size as compared
to solutions based on imposed risk policies determined by management.
Key words: harvest risk; agriculture
History: Received: July 30, 2002; accepted: December 12, 2003. This paper was with the authors 7
1
2
months for
6 revisions.
1. Introduction
Gathering the harvest is a complex managerial prob-
lem in controlling risk. Since the dawn of agriculture
about 12,000 years ago, farmers have dealt with sev-
eral categories of risk in producing the food needed
to sustain their lives and to build urban centers of
industry and learning (Garraty and Gay 1972, p. 50).
For an agricultural cooperative in modern times,
the harvest is the primary source of assets needed to
run the business. Hence, the harvest represents the
base raw material and most important link in the sup-
ply chain for these ?rms. Before each harvest, man-
agers must make critical assessments concerning the
capability of harvesting the entire crop at optimum
maturity.
In this article, we analyze the risk associated with
the harvesting of Concord grapes through the devel-
opment of a mathematical model and the presentation
of results from practice. Speci?cally, the model calcu-
lates the optimal rate for a processing plant to receive
grapes. This is a common situation in agriculture
where growers deliver crops to a central receiving
area for further processing and storage prior to sale.
Determining the optimum rate to harvest and process
the crop is a challenging problem in balancing the
risk of overinvestment in capital with the risk of not
harvesting the entire crop.
2. Structure of the Concord
Grape Industry
The Concord grape, a ?avorful, aromatic purple vari-
ety of grape, is grown in the cooler regions of the
United States. Major growing areas include western
New York, northern Ohio, and northern Pennsylva-
nia (all three near Lake Erie); western Michigan; and
south-central Washington state. In 1869 Thomas B.
Welch, a dentist from New Jersey, used pasteuriza-
tion as a means to preserve Concord grape juice. By
heating the grape juice before bottling he was able to
stop fermentation, creating the ?rst nonalcoholic fruit
juice product. This event marked the beginning of the
bottled and canned fruit juice industry in the United
States.
During the 130-plus years since this genesis, the
market for Concord grapes has grown into a billion-
dollar industry with an annual harvest of about
225
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
226 Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS
400,000 tons. Though the Concord grape comprises
only about 6% of the total grapes grown in the United
States, it maintains a high pro?le because of intensive
marketing; it remains a popular ingredient for juices,
fruit jellies and jams, and concentrates. Not suitable
for the fresh market because of its short shelf life,
nearly all of the Concord grapes harvested are imme-
diately converted into juice.
The leading player of the industry is Welch Foods,
Inc., which is owned by the National Grape Cooper-
ative Association. Welch Foods controls about 55% of
the Concord grapes grown in the United States. The
next-largest player controls less than 7%. Collectively,
?ve business structures comprise the Concord grape
industry.
(1) Private packers that purchase Concord grape
juice as a major ingredient for their products, both
branded and house brands.
(2) Private processors that contract with growers on
an annual basis for delivery of grapes, then process the
grapes into concentrate for sale as a raw ingredient.
(3) Private processors with packaging capabilities, pri-
marily involved in production of house brands.
(4) Cooperatives owned by growers that process
grapes into raw ingredients, sell to other companies,
and return pro?ts to the growers.
(5) Vertically integrated cooperatives involved in
growing, processing, packaging, distributing, and
marketing Concord grape products.
The industry is a mix of both vertically integrated
and modular structures (Fine 1998). However, the
long-term trend is toward supply chain integration
similar to the transition taking place in the poultry
industry (Kinsey 2001). Product life cycles are long,
with few radical product innovations. Intense compe-
tition and lack of pricing power in the market com-
bine to force a focus on cost control.
2.1. Operational Design, Organization, and
Financial Implications
All businesses that convert Concord grapes into
juice—cooperatives and private companies alike—
follow similar steps during the harvest process. In the
fall, usually beginning in late September, fruit grow-
ers deliver grapes picked by mechanical harvesters
to processing plants located near the vineyards. The
grapes are then converted into juice through a com-
plex pressing process. This step takes place within
eight hours after picking to avoid fermentation and
deterioration of quality. Exceeding this time limit
means the grapes must be diverted, at severe cost
penalty, to other uses, such as wine making. The
economics of refrigerated warehousing, as well as
biological degradation at low temperatures, preclude
storage of raw Concord grapes for later pressing into
juice.
In each growing area, approximately 100 mechan-
ical harvesters pick grapes on a continuous basis.
Since the cost of an individual harvester is very
high, approaching $200,000 per unit, growers usually
pool resources through joint ownership or contract-
ing. A single harvester picks an average of three sep-
arately owned vineyards. Coordination of the ?eld
operations becomes critical to ensure a steady feed of
freshly picked grapes to the processing plant.
2.2. Harvest Scheduling and Capacity
Prior to the harvest, planners assign a constant rate of
picking to each harvester operator based on acreage,
estimated yields, the estimated length of the har-
vest, and the standard pressing rate for the receiving
plant. This practice establishes fairness: Each vineyard
owner faces the same risk of losing crop because of
poor weather, such as a frost. A hard frost, where tem-
peratures dip below 28
F, weakens stem tissue, caus-
ing the grapes to fall from the vines. This is called
shelling. Every grower wants to complete the harvest
before a hard frost causes shelling in the vineyards.
The total daily schedule for all harvesters equals
80% of the standard plant pressing rate. The reserve
of 20% allows planners ?exibility during harvest
operations to accommodate individual growers who
might experience crop maturity problems, mechani-
cal breakdowns, or an abnormally high risk of crop
loss due to frost. The reserve is completely used as
part of day-to-day harvest operations so that the rate
of picking always matches the rate of pressing at the
receiving plant. This implies that the rate of picking
and pressing is dictated at the same time that capac-
ity investment decisions are made (i.e., once per year,
projecting ?ve years into the future).
In the midst of harvest operations, it is very dif-
?cult to change the rates of picking and pressing
grapes because of the large number of harvester oper-
ators and the complexity of receiving plant opera-
tions. When day-to-day pressing or picking rates vary
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS 227
by even small amounts a classic system dynamics
response occurs, producing substantial loss of ef?-
ciency that extends many days past the original dis-
ruption. Extensive preharvest efforts focus on testing
and maintenance of plant equipment to ensure relia-
bility in achieving the standard pressing rate per day.
Upon receipt from the vineyards, the grapes are
heated and pressed into unsettled juice that is stored
in large refrigerated tanks. Unsettled juice is the
intermediate step in processing. Welch Foods alone
has over 55 million gallons of tank storage capacity.
Throughout the remainder of the year, the unsettled
juice is further processed and used for bottling and as
an ingredient in other food products. The manufac-
turing structure takes on a “V” shape typical in the
process industries where a small number of raw mate-
rials are used to produce a large number of end items
(Umble 1992). Final processing of the entire harvest
takes close to one year.
2.3. Financial Implications Investment in
Capital Assets
Investment in capital assets to process Concord
grapes into juice is substantial. For cooperatives this
investment in pressing capacity, that remains idle for
10 months per year, is a drain on resources. Despite
tax advantages, cooperatives involved in processing
raw fruits and vegetables typically have a low return
on capital compared with the rest of the food industry.
In this case, the proper sizing of harvest processing
capacity becomes an important strategic decision with
signi?cant ?nancial impacts.
3. Problem De?nition
Consider the typical situation in harvesting Concord
grapes, or other crops, where a planner faces a task
with an uncertain start date, an uncertain end date,
and whose size cannot be known in advance with
certainty. As an example, the maturation dates of
Concord grapes depend on random factors such as
weather and soil conditions as well as cultivation
practices. Random frost dates may signal an abrupt
end to the harvest season and total crop size is not
known in advance of the harvest process.
In such cases, the planner must decide what degree
of effort to devote to the harvest in terms of the press-
ing rate and the corresponding capital investment in
equipment. These are decisions made in advance of
harvest as part of rolling ?ve-year plans because of
the long lead times for capital equipment. A balance
must be made between the costs of the resources com-
mitted and the costs of a partially completed task
(grapes remaining in the vineyard). With whatever
methods employed, the derivation of the optimal har-
vest rate is critical for the grower in controlling risk
and in determining the proper capital investment.
3.1. The Elements of Uncertainty
With uncertainty in both the size of future grape crops
and the time window available for future harvests, it
is impossible to calculate the optimal rate of pressing
using direct methods. Simply dividing the historical
average crop size by an estimate of the available days
of harvest will result in a solution that ignores the
inherent risk of agriculture.
The essence of this problem can be de?ned by three
variables:
H = crop size in tons, a random variable;
| = length of harvest season in days, a random
variable;
í = pressing rate in tons per day, the decision variable.
From these variables two sets of expressions can be
de?ned, one set each for the amount of crop loss (ACL)
and the excess harvest rate (EHR).
ACL =
H ?RL if H >RL
0 if H ?RL
(1)
EHR=
0 if H ?RL
RL?H if H -RL.
(2)
These two sets of expressions have natural interpre-
tations. If H is greater than what can be harvested at
í, then there will be loss in the vineyards, as repre-
sented in Equation (1). This often occurs in practice
when a frost cuts short the harvest season. How-
ever, if í is too high, the opportunity is lost for har-
vesting the crop with less effort, as represented in
Equation (2). This also occurs in practice when there is
overinvestment in pressing equipment. Because both
of the above equations contain two random variables,
we will have to form the expected values of each by
integrating over the joint density function, which we
denote by g(H, |).
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
228 Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS
3.2. Important Model Assumptions
and Justi?cations
Allen and Schuster (2000, pp. 32–33) report that for
one of the Concord grape growing regions a correla-
tion coef?cient of 0.14 exists between H and | based
on 22 years of data at an observed signi?cance of
53%. The results of this analysis offer support that H
and | are statistically independent. This makes sense
because H is dependent on many variables such as
precipitation, soil conditions, temperatures, and avail-
able sunlight, which are uncorrelated to the time win-
dow of good weather available to gather the harvest.
Then the joint density function can be formed as the
product of two marginal distributions.
g(H, |) =] (H) ×] (|). (3)
Another important assumption is the shape of
the probability distributions used to estimate H and
|. Extensive historical information from state agri-
cultural experiment stations, spanning more than
40 years of history, allows calculation of the mean
and standard deviation in each case. Histograms
of this data show that both H and | are mound
shaped and reasonably symmetrical. An analysis of
data from Allen and Schuster (2000, pp. 32–33), using
a chi-square goodness-of-?t test, shows insuf?cient
evidence (p = 0.53 and p = 0.66) to reject the null
hypothesis that H and | are normally distributed. In
both cases, one degree of freedom exists. The modi-
?ed histograms of H and | contain four observations
in one and two intervals, respectively. This is out-
side the minimum guideline of ?ve observations for
the chi-square goodness-of-?t test. It is also impor-
tant to note that in similar situations researchers use
symmetrical distributions (Jones et al. 2001). These
observations lead us to assume | and H are normally
distributed with mean and standard deviation j
|
, u
|
and j
H
, u
H
respectively.
3.3. The Objective Function
The Harvest Model minimizes a total cost function
that contains two costs.
• Underage cost (C
H
, dollars per ton). This re?ects the
cost of harvesting too slowly, therefore foregoing sales
of grapes left in the ?eld at the end of the harvest
season.
• Overage cost (C
í
, dollars per ton). This re?ects the
cost of harvesting too quickly, therefore committing
an excessive amount of resources to the harvesting
effort.
The formation of the Harvest Model involves
applying costs to ACL and EHR. Because the Har-
vest Model is probabilistic in nature, expected values
are needed for both ACL and EHR. Once expressions
exist for each, costs can be applied to form a total cost
function. The minimum of this function will provide
the optimal harvest rate at minimum cost.
The remainder of this study will focus on the mat-
ter of harvesting Concord grapes with an uncertain
length of harvest season and an uncertain crop size.
Section 4 provides an overview of the literature relat-
ing to the Harvest Model. Section 5 gives speci?cs
about the derivation of the expected values and the
minimization of the total cost function. In §6 detailed
numerical results of the Harvest Model are presented
based on real data, followed by §7, which provides
managerial insights.
4. A Review of Cognate Research
The analysis of harvest risk draws upon three gen-
eral areas of previous research, including the analysis
of harvesting operations, agricultural decision making
involving risk, and the newsvendor model.
4.1. Analysis of Harvest Operations
Thornthwaite (1953) addresses the problem of har-
vesting peas at the peak of maturity. Subsequent
analysis by Kreiner (1994) shows that Thornthwaite’s
method was still in use more than 40 years later
and has reduced harvest and processing costs, and
lost product. Though Thornthwaite’s work involves
annual crops (peas), and does not deal with the prob-
ability of losing crop as a result of frost, it remains
a classic in the application of operations research. To
our knowledge this is the ?rst reference on the scien-
ti?c study of harvest operations.
Porteus (1993a, 1993b) develops a two-part case
study of the National Cranberry Cooperative. In this
case, he examines complex trade-offs in capital invest-
ment and capacity for the processing of cranber-
ries during a harvest season. Although the harvest
and processing of cranberries share common elements
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS 229
with Concord grapes, the author did not address the
risk of crop loss from a frost, or from overmaturation.
In aggregate, these four references provide impor-
tant background information on harvesting opera-
tions plus speci?cs about the uniqueness of this area
of management.
4.2. Agricultural Decision Making
Involving Risk
Maatman et al. (2002) note that “various methods
of risk reduction exist, including those aimed at pre-
vention of risk (e.g., irrigation), dispersion of risk (by
diversi?cation of risky activities such as the cultiva-
tion of different varieties of crops), control of risk (e.g.,
by sequential decision making) and ‘insurance’ against
risk” (p. 400, italics in original).
Much of research in agricultural decision making
focuses on the dispersion of risk through improved
individual farm planning. Mathematical program-
ming is often the tool used for modeling farmers’
strategies under uncertain conditions (see Anderson
et al. 1977, Hazell and Norton 1986). These models are
static in nature, assuming that probabilities of occur-
rence are known in advance and that all decisions are
made at one time.
This class of models is similar in spirit to what is
presented in this article. However, the Harvest Model
does not focus on the dispersion of economic risk and
does not employ traditional mathematical program-
ming. Rather, we concentrate on control of risk for a
pooled harvest situation (a cooperative) through opti-
mization based on a generalization of the newsvendor
problem.
Jones et al. (2001) study the control of risk for pro-
duction of hybrid seed corn from the perspective of
both random yields and demands. A follow-up study
reports results in practice (Jones et al. 2003). The
authors’ work is similar in style to the Harvest Model.
However, the authors do not address uncertain start
and end times for a harvest, choosing instead to focus
on a two-stage production model that involves two
different regions (North America and South America)
with staggered growing seasons. In this regard, their
work differs from the Harvest Model, and will no
doubt lead to further work in controlling agricultural
risk.
Previous research involving Concord grapes and
agricultural cooperatives deals with decision making
in raw materials management (Schuster and Allen
1998, Schuster et al. 2000) and production planning
(Schuster and Finch 1990, Allen and Schuster 1994,
Allen et al. 1997, D’Itri et al. 1998). Though these mod-
els yield signi?cant ?nancial savings in practice, none
considers the risk associated with harvesting Concord
grapes.
In a recent article, Allen and Schuster (2000) add-
ress harvest risk for Concord grapes through a work-
ing model with the successful outcome of a closed
form solution. Using this approach, senior manage-
ment speci?es risk levels as policy. An example would
be “Harvest 100% of the crop, 85% of the time.”
The policy level becomes an important input. Despite
offering a large improvement over previous methods
that involved heuristics, this model does not include
an explicit treatment of the cost of lost crop or sea-
sonal operating costs.
4.3. The Newsvendor Model
The newsvendor approach is effective in a num-
ber of applications, including the defense industry
(Masters 1987), and is the underpinning of our har-
vest model. Nahmias (1997, pp. 272–280) provides
a comprehensive review of the newsvendor model
including derivation, historical context, and several
important extensions.
The process of harvesting grapes shares character-
istics with the management of style goods such as
fashion items in the clothing industry. As is true of
style goods, grapes are perishable. Concord grapes
have a limited time for peak maturity combined with
a short time window for harvesting during the fall. In
this regard, the harvest of Concord grapes shares sim-
ilarities to seasonal sales of clothing where there are
substantial costs of obsolescence and limited oppor-
tunities to store for future sale.
An important body of literature addresses the style
goods problem. Fisher and Raman (1996) develop a
response-based production strategy to deal with the
uncertain demand characteristic of style goods. Other
authors improve style goods forecasts by using a
Bayesian approach (Eppen and Iyer 1997, Murry and
Silver 1966). However, with the harvest of Concord
grapes, little opportunity exists to change the harvest
rate during seasonal operations, rendering Bayesian
methods less effective.
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
230 Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS
As a concluding comment, the distinguishing
aspect of the Harvest Model is that it contains two
random variables (H and |) as compared to the tradi-
tional newsvendor model that contains only a single
random variable (demand). In this way, the harvest
risk model shares similarities with the work of Song
et al. (2000) that considers two random variables.
5. Mathematical Development
We now turn our attention to expressions for the
expected ACL and EHR.
5.1. The Expected ACL
Since the ACL is discontinuous at H =RL, the expec-
ted value is given by
E(ACL) =
|=
|=?
H =
H =RL
(H ?RL)] (H) dH
] (|) d|.
(4)
The inner integral is the familiar normal loss function.
We de?ne the standard normal variate for H as
z
H
=
H ?j
H
u
H
, (5)
and
Q=
RL?j
H
u
H
. (6)
Then the expected ACL becomes
E(ACL) =u
H
?
G(Q)] (|) d| (7)
where
G(Q) =4(Q) ?Q|1 ?+(Q)] (8)
and 4(·) is the standard normal density function
while +(·) is the cumulative of the standard normal
density function. The form of the expected ACL can
be simpli?ed by introducing the nondimensional har-
vest rate, r. In a risk-free environment, the risk-free
harvest rate becomes í
0
= j
H
,j
|
. In this situation,
í is known because both H and | are known in
advance. Of course in reality uncertain crop size and
uncertain harvest start and end times exist, so í is
different from í
0
. The de?nition for the nondimen-
sional harvest rate becomes r =í,í
0
. Next, we de?ne
the standard normal variate for | as
z
|
=
|?j
|
u
|
. (9)
Finally, we de?ne the two coef?cients of variation
|
H
=u
H
,j
H
, |
|
=u
|
,j
|
.
Now Q can be expressed entirely in terms of the two
coef?cients of variation, the standard variate, z
|
, and
the dimensionless harvest rate, r:
Q=
r(1 +|
|
z
|
) ?1
|
H
. (10)
Then the expected ACL becomes
E(ACL) =j
H
|
H
?
G(Q)4(z
|
) dz
|
. (11)
Silver and Smith (1981) provide the tools to carry
out the integration of this expression and subsequent
results. First note that Q in Equation (10) can be
expressed in the form Q=uz
|
+|, where
u =r(|
|
,|
H
), | =(r ?1),|
H
. (12)
In the interest of notational conciseness, we de?ne
another recurring parameter as
c =|,
1 +u
2
. (13)
Using Equations (12) and (13), the integrated form of
Equation (11) becomes
E(ACL) =j
H
|
H
1 +u
2
G(c). (14)
In Equation (14) and all subsequent results, G(·) is
de?ned by Equation (8) and we emphasize that 4(·) is
the standard normal density function (i.e., zero mean
and unit variance) and +(·) is the cumulative of the
standard normal density function.
5.2. Expected EHR
Similar to ACL, this function is also discontinuous at
H =RL so
E(EHR) =
|=
|=?
H =RL
H =?
(RL?H)] (H)] (|) dH d|.
Now add and subtract the complement to the inner
integral and rearrange to get
E(EHR) =
|=
|=?
u
H
G(Q) +
H =
H =?
(RL?H)] (H) dH
· ] (|) d|,
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS 231
where Q is as previously de?ned in Equation (10).
The inner integral simpli?es to |RL?j
H
] and on sub-
sequent integration over | this becomes |íj
|
?j
H
].
This should be in nondimensional form as well, so
using earlier de?nitions, the result is
E(EHR) =j
H
|
H
?
G(Q)4(z
|
) dz
|
+j
H
(r ?1). (15)
The integrated form of Equation (15) becomes
E(EHR) =j
H
|
H
1 +u
2
G(c) +j
H
(r ?1). (16)
5.3. Expected Total Cost and Optimal Harvest Rate
The integral form of the total cost (TCP) per ton of
harvest can now be expressed as an expected value
using Equations (11) and (15):
E(TCF),j
H
= C
H
|
H
?
G(Q)4(z
|
) dz
|
+C
í
|
H
?
G(Q)4(z
|
) dz
|
+C
í
(r ?1). (17)
The integrated form of Equation (17) is
E(TCF),j
H
= (C
í
+C
H
)|
H
1 +u
2
G(c)
+C
í
(r ?1). (18)
Next, an expression is obtained for the optimal
dimensionless harvest rate. Taking the ?rst derivative
of Equation (17) with respect to r, setting the result to
zero, and rearranging terms yields an implicit expres-
sion for the optimal dimensionless harvest rate (r
?
):
?
|1 ?+(Q)](1 +|
|
z
|
)4(z
|
) dz
|
=C
í
,(C
í
+C
H
). (19)
Again using Silver’s results, the integrated form of
Equation (19) yields
+(c) +|
|
u,
1 +u
2
4(c) =1 ?C
í
,(C
í
+C
H
), (20)
where u and c are functions of the decision variable r
and the parameters |
H
and |
|
through Equation (12).
It deserves emphasis that r
?
depends only on the
nondimensional ratios |
H
, |
|
; and the cost ratio C
í
,
(C
í
+ C
H
). Furthermore, the second derivative of
Equation (17) with respect to r is positive for all pos-
sible z values so that Equation (20) will yield the min-
imum r.
5.4. Expected Percentage Crop Recovery
Beyond r
?
, the Harvest Model also permits calcula-
tion of the expected amount of crop recovery. As men-
tionedpreviously, the introductionof riskintoplanning
means that the harvest will be less than 100%.
We de?ne the expected percentage crop recovery by
E(PCR) = |expected amount of crop harvested
/mean harvest size] ×100.
The expected proportion of ACL is E(ACL)/j
H
. Then
E(PCR) =|1 ?E(ACL),j
H
] ×100. (21)
5.5. An Alternative Approach to Harvest Risk
Management
Some decision makers prefer a policy approach (as
opposed to a cost-based method) for dealing with risk
management. In an earlier study, Allen and Schuster
(2000) examine a policy approach. The objective here
is to explore the connections between that study and
the cost-based analysis in the present study. In this
alternative, instead of specifying cost penalties man-
agement speci?es a policy for the probability of har-
vesting the entire crop. It is important to emphasize
that this is not at all the same as specifying the pro-
portion of crop harvested as was addressed in the
previous section.
The probability of harvesting the entire crop is
found from
Prob(complete crop harvest)
?P(CCH) =P(H ?RL) =1 ?P(H >RL)
P(CCH) =1 ?
|=
|=?
H =
H =RL
] (H) dH
] (|) d|.
The inner integral is simply |1?í (RL)], so again using
the de?nition of Q in Equation (10) employed in our
earlier work, we obtain
P(CCH) =1 ?
?
|1 ?+(Q)]4(z
|
) dz
|
.
We will illustrate the use of this policy-based risk
management method for the case of a speci?ed
P(CCH) =0.85 (85%) in §6.2. Integration of the above
yields the following implicit equation for the dimen-
sionless harvest rate, r:
+(c) =P(CCH) =0.85. (22)
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
232 Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS
6. Numerical Results
This section draws from real data on the harvest of
Concord grapes documented by Allen and Schuster
(2000) as a basis for examples and analysis. Their
work deals speci?cally with the decision-making pro-
cess involving the harvest of Concord grapes for
Welch Foods. The authors were directly involved
in planning and managing harvest operations for a
period of 10 years.
6.1. Data
The model requires two sets of historical data: H
(tons) and | (days). Both H and | vary from year
to year. In each growing area, the state agricultural
experiment station keeps detailed records by year on
yields and crop size. Historical records also show the
harvest start date (when the crop is at the correct level
of maturity) and the ?rst 28
F day (hard frost) that
signi?es the end of the harvest. The value of | for
each harvest can be calculated for a growing region
by counting the days between the start and end dates.
Similar to the crop size, | varies from year to year.
Besides H and |, the Harvest Model requires two
costs, C
H
and C
í
. The calculation of C
H
is straightfor-
ward; if a grower does not harvest a ton of grapes,
then the relative value of the loss equals the open
market price. This is a conservative approach to val-
uation. The examples presented assume C
H
= $250
per ton of grapes. This is a long-term estimate of the
open market price. The market price of grapes will
change a considerable amount on a year-to-year basis
depending on industry crop size, carryover from the
previous year, consumer demand, and the pricing of
alternative raw ingredients.
The calculation of C
í
is a more complex matter.
Harvesting at an excessive rate means that the entire
harvest will take place well in advance of a hard frost.
If this happens, then í is too high and the long-term
investment in capital is excessive. In this situation C
í
relates to the amortized cost of the unneeded capacity.
It is unique to each type of business organization.
Estimating C
í
is dif?cult because it involves capital
equipment that is added in steps. There is no question
that the cost is nonlinear over large ranges of í. How-
ever, for small changes of í the cost becomes linear.
An estimate can be calculated on a cost-per-ton basis
by analyzing small increases of í, e.g., an increase of
10% or less and the corresponding capital required to
support the higher pressing rate. This was the proce-
dure used at Welch Foods.
For example, at one grape processing plant located
in Pennsylvania, increasing í by 180 tons per day
requires a capital investment of $1.5 million. During
the course of an average harvest, this increased rate
means that 5,400 extra tons of grapes can be pressed
before a hard frost. Assuming a 10 year life span,
the estimated amortized cost of this extra capacity is
$150,000 per year. If part or all of this extra capacity is
not needed, then C
í
becomes the cost of the incremen-
tal capacity divided by the additional tons harvested
at the higher rate ($150,000 per 5,400 tons), or about
$30 per ton of grapes. This represents the estimated cost
of incremental capacity occurring around the point in
time when a hard frost might occur.
6.2. Results
We now provide numerical results from the Harvest
Model for the following practical problems:
• calculation of r
?
under different scenarios,
• determination of the percent recovery for a spe-
ci?c r
?
, and
• ?nding r under conditions of an imposed harvest
policy.
Using the historical data for H and | along with
C
H
and C
í
as inputs, the Harvest Model calculates
r
?
providing the opportunity to analyze a number of
harvest risk scenarios.
Table 1 gives results for r
?
of various combinations
of |
H
and |
|
for a cost ratio, C
í
,(C
í
+ C
H
), of 0.10
using Equation (20). These were obtained by using the
Goal Seek utility in Excel with a tolerance of 0.0001 for
all calculations of r. The data show that r
?
increases
with increases in variability for H and |. This is a con-
sistent result. The greater the risk, the greater the r
?
.
Table 1 Optimal r for C
R
/C
R
+C
H
=01
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 13350 13564 13828 14195 14620
030 13880 14019 14291 14618 15004
035 14306 14449 14662 14954 15305
040 14586 14732 14879 15167 15489
045 14713 14860 15008 15239 15532
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS 233
Figure 1 Cost Ratio vs. r
?
for k
H
=025
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Optimal Dimensionless Harvest Rate, r*
C
o
s
t
R
a
t
i
o
C
R
/
(
C
R
+
C
H
)
k
L
=0.05
k
L
=0.15
k
L
=0.25
k
L
=0.35
k
L
=0.45
Additional descriptive results can be observed by
evaluating the left-hand side of Equation (20) over
ranges of r
?
for representative values of |
H
and |
|
.
The resulting values correspond to the cost ratios at
optimality. The set of graphs in Figure 1 are for a con-
stant |
H
of 0.25 and a range of |
|
values. An estimate
of r
?
can be made directly from the graph without
invoking a search engine by specifying the cost ratio
and the coef?cient of variation for the length of har-
vest season. This provides planners with a visualiza-
tion of the relationship between the cost ratio and r
?
.
Computations of expected percentage of crop
recovery using Equations (14) and (21) for the r
?
of
Table 1 (for C
í
,(C
í
+C
H
) = 0.10), appear in Table 2
below.
Increasing variability in crop size and length of season
means lower crop recovery. This re?ects the trade-off
Table 2 E(PCR) in %
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 9716 9708 9695 9679 9661
030 9596 9591 9580 9565 9548
035 9452 9446 9435 9421 9404
040 9275 9271 9256 9244 9227
045 9067 9063 9049 9035 9018
Table 3 Total Cost per Harvest Ton at Optimal r
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 1731 1804 1919 2066 2237
030 2204 2268 2369 2501 2659
035 2728 2785 2876 2997 3144
040 3300 3352 3436 3548 3686
045 3913 3962 4041 4147 4278
between the cost of unharvested crop and the cost
of an EHR. Table 2 is an important tool to explain
why 100% recovery of a crop is not optimal given
the trade-off between crop loss and a consistent early
completion of harvest operations.
The total cost per ton for each value of r
?
can be
calculated using Equation (18). These results appear
in Table 3. As is the case with r
?
, the data show that
cost per ton increases with increases in variability for
H and |.
Finally, Goal Seek can be used to ?nd r for speci?ed
values of |
H
and |
|
under an imposed policy for crop
recovery, Equation (22). Decision makers often use
this approach to demonstrate how r changes for pol-
icy alternatives (suboptimal values of r) approaching
a complete harvest. Since a policy alternative might
not be optimal, it is important to understand the cost
difference between the optimal solution and the cho-
sen policy. In some cases, choosing an r that is not
optimal is justi?ed based on the timing of capacity
additions in relation to legacy equipment, an estab-
lished consensus by growers who are risk adverse,
and the uncertainty surrounding cost data or a shift in
historical trends. By substituting the values of r into
Equation (18) total costs can be computed for each
imposed policy alternative. These results appear in
Tables 4 and 5. The imposed policy for this example
is “Harvest 100% of the crop 85% of the time.”
Table 4 Policy r for 85%
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 13547 13682 13927 14225 14573
030 14570 14684 14883 15144 15458
035 15729 15842 16001 16249 16533
040 17094 17265 17371 17545 17837
045 18716 18903 19002 19192 19384
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
234 Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS
Table 5 Total Cost in US$ per Harvest Ton for Policy r
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 1733 1805 1919 2066 2238
030 2226 2286 2384 2513 2667
035 2798 2852 2936 3051 3191
040 3472 3528 3598 3696 3827
045 4275 4332 4394 4487 4596
Table 6 shows the cost penalties for the 85% policy
as compared to r
?
for each case of |
|
and |
H
. The cost
penalty for the 85% policy can be computed using the
following formula:
percent cost penalty = (policy cost–optimal cost)
· 100,optimal cost.
At the extreme (|
|
= 0.45), the maximum cost
penalty for the 85% policy is about 9.2%. For lower
values of |
|
, the penalty is less than 3%. It is inter-
esting to note that in all cases for a given |
|
the cost
penalty decreases as |
H
increases. The greater vari-
ability in crop size has the impact of narrowing the
cost difference between r, at 85% policy, and r
?
. Since
in all cases the penalties are modest, we conclude that
the 85% policy gives an r that is close to optimum in
terms of cost (for the set of parameters examined).
The Harvest Model also provides results based on
various policy alternatives that are close to those
obtained in the study by Allen and Schuster (2000)
with some small differences in the second decimal
place for the lower values of |
H
. The more accurate
values are probably those of the prior study that did
not rely on Goal Seek. This validates the Harvest
Model. The above results are not optimum from the
point of view of minimizing costs but rather are the
result of an imposed policy. This is an effective way
of communicating harvest risk to growers investing
in receiving plants through agricultural cooperatives.
Table 6 Cost Penalty for Policy r in %
k
H
0.05 0.1 0.15 0.2 0.25
k
L
025 014 007 003 000 000
030 096 081 063 046 031
035 256 239 207 181 150
040 522 525 472 417 382
045 923 934 874 819 743
The Harvest Model gives consistent answers to a
number of important questions involving risk. Based
upon considerable use in practice, the Harvest Model
is suitable as a general solution for problems encoun-
tered in agriculture where a sudden event, such as a
fall frost, concludes the harvest season.
7. Managerial Insights
The true test of any model is performance in practice.
The Harvest Model consistently provides reasonable
results in addition to new insights. As an example,
the traditional belief that the Concord harvest sea-
son should last 30 days to ensure complete harvest
of the crop proves inaccurate when analyzed using
the Harvest Model. Regional climatic variation plays
a signi?cant role in the probability of a frost. A large
body of water, such as Lake Erie, acts as a heat sink in
the fall, thereby extending the probable length of the
harvest season for growers in close proximity. Anal-
ysis using the Harvest Model shows that extending
seasonal operations beyond 30 days in parts of the
Lake Erie region can be accomplished with low risk
of crop loss. This translates into a slower harvest rate
and reduced capital investment for cooperatives in
this growing area.
However, similar analysis for growers in Michigan,
who are located a considerable distance from any
large body of water, shows that seasonal operations
should be less than 30 days because of signi?cant frost
risk. In this growing region a small probability exists
that a frost will occur before the beginning of the har-
vest. Given this case, the harvest rate must increase,
which requires more capital investment.
The authors have experience implementing results
of the Harvest Model, using normal probability dis-
tributions to express harvest size and length of sea-
son, for growing areas near Lakes Erie and Michigan.
In total, capital avoidance equaled $2 million through
establishing the proper harvest rate by region. This
is one example of the bene?ts of controlling harvest
risk, and of the adoption of precision agriculture in
practice.
Beyond savings in capital, the Harvest Model pro-
vides other organizational bene?ts. In the case of
Welch Foods, the Harvest Model is an important
tool for corporate planning with results of the model
being presented at meetings of the board of directors.
Allen and Schuster: Controlling the Risk for an Agricultural Harvest
Manufacturing & Service Operations Management 6(3), pp. 225–236, ©2004 INFORMS 235
Welch Foods is owned by the National Grape Coop-
erative and is an example of vertical integration in
agriculture. The company’s span of control ranges
from raw grapes to end-product marketing. A critical
measure of effectiveness is the capability of receiving
the grower’s grapes with little risk of loss. By using
the Harvest Model, management can communicate to
growers—the owners of the cooperative—exact levels
of risk associated with harvesting the crop. Shifting
dialogue to a fact-based discussion offers great advan-
tages in reaching a consensus concerning the proper
investment in capital for pressing operations.
In addition, the Harvest Model provides a struc-
ture to evaluate the impact on picking and pressing
operations from additions of vineyards to a coopera-
tive. As product lines expand and sales grow, the need
for Concord grapes also increases, causing additional
burden in harvesting. The decision to add incremental
acreage to the cooperative is in?uenced by the capa-
bility to harvest the crop at acceptable levels of risk
and capital investment. The Harvest Model provides
an effective structure to evaluate the change in risk
when additional vineyards are planted. Since risk lev-
els vary by growing region, the proper decision can
be made concerning the correct area to plant addi-
tional grapes. This is a safeguard against expansion
in high-risk growing regions where the probability of
a killer frost is high. Over the long run, this type of
prudent, risk-based decision making will result in an
increased level of reliability for raw material supply.
8. Conclusion
We live in a world ?lled with quanti?able risk, yet
there are few mathematical frameworks available to
guide practitioners who must deal with agricultural
harvest risk. Based on an extensive literature search,
we conclude that harvest risk is an undeveloped
area within operations management. Controlling risk
by the application of rational models presents many
opportunities to increase pro?ts.
In this article, we develop a robust mathematical
model to optimize the risk of an agricultural harvest.
We test the model with representative data from the
harvest of Concord grapes and ?nd it gives accept-
able solutions. During the course of our work we
also ?nd the model applies to other risk-bearing tasks
in agriculture, such as the harvest of oranges, and
in industries such as project management and style
goods planning. We plan further research in this area.
Acknowledgments
The authors wish to thank the editor, senior editor, and two
anonymous reviewers for the constructive comments pro-
vided during the review process.
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