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ABSTRACT *
Loss of dairy delivery cases continues to plague Florida dairy
processors despite control measures adopted by many firms in recent
years. Losses to Florida processors in 1981 may exceed $2,000,000.
This study provides managers of dairy processing firms with a
management tool which will help to (1) determine the rate of loss for
different types of cases and (2) determine loss rates by age of case.
Key words: dairy delivery cases, dairy processing, dairy delivery
case loss rates
Determining Dairy Delivery Case Life Using
a Practical Sampling Procedure
Robert L. Degner and J. Scott Shonkwiler
September 1981
The Florida Agricultural Market Research Center
a part of
The Food and Resource Economics Department
Institute of Food and Agricultural Sciences
University of Florida, Gainesville, Florida
The Florida Agricultural Mrket Research Center
A Service of
The Food and Resource Economics Department
Institute of Food and Agricultural Sciences
The purpose of this Center is to provide timely, applied research
on current and emerging marketing problems affecting Florida's Agri
cultural and marine industries. The Center seeks to provide research
and information to production, marketing, and processing firms, groups
and organizations concerned with improving and expanding markets for Florida
agricultural and marine products.
The Center is staffed by a basic group of economists trained in
agriculture and marketing. In addition, cooperating personnel from IFAS
units provide a wide range of expertise which can be applied as determined
by the requirements of individual projects.
TABLE OF CONTENTS '
Page
LIST OF TABLES............................ ................ ..... iv
LIST OF APPENDIX TABLE....... ..... .............. ..... ............. iv
LIST OF FIGURES.................................................. iv
LIST OF APPENDIX FIGURE............ .....................,....... iv
ACKNOWLEDGEMENTS.................................................. v
SUMMARY................ .. .. .............. ............. ..... .... .. vi
OBJECTIVES............................................. ..... .... 2
RESEARCH PROCEDURE............... ................................... 2
Determining Case Life Through Sampling ........................ 3
Obtaining the Sample................................ ... ... 6
Analyzing the Sample: An Example.............................. 8
CONCLUSIONS....................................................... 12
APPENDIX................................ .......................... 13
REFERENCES................................... .... ................. 18
iii
LIST OF TABLES
Table
1 Sampling rates for various types of cases or types of dates.....
2 An example of data and calculations required to determine case
life..................................... ........................
Page
4
9
LIST OF APPENDIX TABLE
Page
Table
1 An example of data and calculations required to determine
case life using a statistical procedure........................
LIST OF FIGURES
Figure
1 Typical date codes found on dairy delivery cases. Both
indicate a case manufactured in March of 1981.................
2 Percent of cases remaining in a distribution system over time
(from Table 2, Column 6) ....................................
Page
LIST OF APPENDIX FIGURE
Figure
1 Percent of cases remaining in a distribution system over time...
Page
17
ACKNOWLEDGEMENTS
This research was funded in part by a grant from the Florida Dairy
Products Association. Mr. R. Gene Smith, Chairman of the Board, and
Mr. Joseph R. Antink, President, gave leadership to the Association in
supporting this project.
Our appreciation is expressed to executives and employees of Borden,
Inc., Orlando and Tallahassee, T. G. Lee Foods, Orlando, and Pet, Inc.,
St. Petersburg, for their excellent cooperation in providing data and
access to their plants for developing and testing the case sampling
procedure.
Thanks are also due Ms. Patricia Beville and Mrs. Lois Schoen for
typing the manuscript and Mrs. Susan Howard for preparing the graphs,
and to Mtss Judy King for data processing.
SUMMARY
* Loss of dairy delivery cases continues to plague Florida dairy
processors despite control measures adopted by many firms in recent
years. Losses to Florida processors in 1981 may exceed $2,000,000.
* This study provides managers of dairy processing firms with a
management tool which will help to (1) determine the rate of loss
for different types of cases and (2) determine loss rates by age
of case.
* A statistical sampling technique and analytical method has been
developed which is simple and inexpensive to administer.
* The technique requires that each batch of cases be uniquely identifi
able; the most practical way of doing this is to have the manufacturer
imprint each lot of cases with a manufacture date.
* The dates of service for each batch of cases must also be ascertain
able from company records.
* Large, legible manufacture dates facilitate the sampling procedure.
The "easiest to read" dates can be sampled at the rate of 1,4001,500
per hour compared with only 200300 per hour for difficult to read
date codes.
* The recommended sample size is 2,000 cases.
* Sampling may be done at any time after the most recent batch of cases
has been thoroughly distributed throughout the system, usually about
one month after introduction.
* Sampling may be done at any convenient location in the case storage
area or online. If done online, an acceptable procedure would
be to select every "nth" case as it passes by.
* Avoid bias in drawing the sample. Take care to include every case
selected, regardless of type or legibility of date. Every effort
must be made to determine dates of all cases selected.
* Tabulate the number of cases observed for each manufacture date. Then
determine the proportion of each date to the original number purchased
and placed into service. This proportion or ratio is the "sample
ratio".
* The sample ratio for the most recent batch of cases in the system
serves as a base. If it is assumed that none of the newest cases is
missing, the sample ratio for the most recent batch is one, or
equivalent to 100 percent (the percent of cases still in the system).
The sample ratios for the other case dates expressed as a ratio to
the sample ratio base reflects the proportion of each date left in
the system.
* Comparison of results for different company locations, case types,
or management practices can help identify ways to reduce case costs.
Determining Dairy Delivery Case Life Using a
Sampling Procedu're
Robert L. Degner and J. Scott Shonkwiler
Studies conducted by the Florida Agricultural Market Research
Center (FAMRC) in 1977 and 1979 placed dairy delivery case losses for
Florida dairy processors during 1976 and 1978 at $1.3 and $1.5 million,
respectively (Mathis and Degner, 1977, 1979). Case losses cost pro
cessors 0.53 and 0.56 cents per gallon of fluid product for the re
spective study years.
Although most Florida dairy processing firms have adopted measures
to ameliorate the case loss problem, many plant managers feel that
losses are greater than ever. If the 1978 costs are simply adjusted for
inflation, the current annual losses exceed $2 million. This continuing
problem and the associated escalating costs make it imperative that
processing plant managers be able to accurately monitor their case
inventories and estimate the rate at which cases disappear from their
distribution systems. Knowing the disappearance rate for cases of
different materials or design, different case management policies, or
for individual plants, would provide a management tool which could
reduce future case losses. Further, accurate information on loss rates
may be used to modify depreciation schedules, possibly resulting in tax
savings.
OBJECTIVES
The primary objective of this study was to develop a technique for
determining the rate of loss of dairy delivery cases for individual
firms. Specific objectives were to:
1. Determine the relative loss rates for cases made of wire and
plastic.
2. Determine case loss rates by age of case.
3. Provide processors with a sampling method to meet the above
objectives which is relatively simple and inexpensive to
administer.
RESEARCH PROCEDURE
The theoretical basis for the study rests on the ability to compare
an observed (sample) distribution of dairy cases remaining in a system
with a known distribution (over time) of case acquisitions. This com
parison requires 1) the ability to determine the actual numbers of dairy
delivery cases placed in service by a given processing plant at specific
points in time (from processing plant records) and 2) the ability to
identify individual cases as to the date placed into service (from manu
facture dates or codes stamped on cases).
Four major processing plants cooperated in the study. All provided
case purchase records for a minimum of two years and all allowed access
to case handling areas so that researchers could observe case flow rates
and record case manufacture dates from a sample of cases. All plants
were visited by FAMRC personnel one or more times in early 1981.
Unfortunately, none of the cooperating plants maintained the numbers
of cases of a given manufacture date put into service on specific dates.
Several plants had also received shipments of cases which had no manu
facture date, thereby precluding the use oi their case data. However,
it was possible to examine records of case purchases and relate these to
manufacture dates found on cases in several plants so as to obtain
sufficient data to develop the basic procedure. Further, information on
time requirements and other practical aspects of sampling was obtained
at all cooperating plants. Several techniques were developed and evaluat
ed, but the one described in the following section was judged to be the
simplest and most practical to use, and yet very effective.
Determining Case Life Through Sampling
In order to use the case sampling procedure described below, a pro
cessing plant must have 1) cases which have manufacture dates or codes
imprinted on them and 2) accurate records which indicate when cases of a
particular manufacture date were put into service.
The case manufacture date or code is extremely important because it
uniquely identifies a given batch of cases. Most case manufacturers
routinely imprint or emboss numbers representing the month and year on
their cases. On plastic cases the numerals are usually found imprinted
on two opposing side panels. On wire cases, they are usually stamped or
embossed on several of the sheet metal corner supports.
Some cases, particularly those made of plastic, will have a date
code rather than a numeric month and year date. Typically, such codes
will consist of a grid or "clock" which will contain one dot for January,
two for February, etc. The year of manufacture will usually be found in
the center of the grid or circle (Figure 1). On some cases the code is
found on a side panel, and on others, the bottom. Obviously, the size
and location of the manufacture date greatly affects legibility, which
in turn affects the time required to record dates from a sample of
cases. Plastic cases with relatively large ( inch to 1 inch) numerals
in contrasting colors on side panels could be sampled at the rate of
1,400 to 1,500 per hour. Metal cases with numerals about the same size
were sampled at 400 to 500 per hour, and only 200 to 300 plastic cases
with date codes could be sampled per hour (Table 1).
Table 1.Sampling rates for various types of cases or types of dates.
Type of case, date Sampling rate
(Cases per hour)
Plastic, large contrasting numerals 1400 1500
Metal, large embossed numerals 400 500
Plastic, coded dates 200 300
Obviously, the sampling rate depends on the skill and experience of
the sampler, the location in the plant where sampling is done, and the
method used to record the datj. FAMRC researchers found that dictating
dates into a portable tape re order saved considerable time compared
with writing them down. If it is anticipated that the dates will be
used for sampling purposes, every effort should be made to get the
manufacturer to put large numeric dates on the side panels or the corner
supports. Codes should be avoided, as they require considerable more
time to decipher and record.
0 00
81
Note: actual size
is approximately
one inch.
Figure 1. Typical date codes found on dairy delivery
cases. Both indicate a case manufactured
in March of 1981.
Most plants visited had an assortment of different case types, with
different types of dates or codes. When sampling from such an assortment,
it is important to read every date on cases included in the sample, not
just those that are most legible. This will reduce sample bias, which
is discussed in greater detail in the next section. If the above con
ditions have been met, i.e., dated cases and accurate records indicating
when cases of a particular manufacture date were put into service, this
procedure can be used. The next step is to record manufacture dates
from a random sample of cases.
Obtaining the Sample
The optimum time for sampling cases is as soon as possible after
the most recent batch of cases has been absorbed by the system and has
become randomly distributed. While the exact time is difficult to
specify, it is estimated to be about one month after introduction of new
cases to the system for most plants. If sampling is done too soon, the
majority of the newest cases might remain together, and could be under
sampled if they are all at customers or away from the plant at once.
Similarly, they could be oversampled if large numbers are back at the
plant simultaneously. On the other hand, if too much time elapses after
introduction of a batch of cases into the system, substantial numbers
could already have been lost and estimates of case losses based on the
technique described in the next section may be too low.
A sample of approximately 2,000 cases is sufficient for most plants
to insure reasonable accuracy. Every effort must be made to insure that
a random sample of dates will be recorded. In most plants, cases of all
ages and types will be uniformly or randomly distributed through the
distribution system. However, if cases tend to be sorted for some
reason, i.e., because of driver preference for a particular type of case
or because some cases pose handling problems in certain lines, care must
be exercised or a biased sample may result.
At the four cooperating plants, samples were obtained in various
locations such as filling lines, case storage areas, unloading docks,
and the case washing line. The sample dates should be obtained wherever
dates can be easily read, without introducing bias into the sampling
procedure.
Bias can be introduced whenever the sampler cannot read the date on
every case that passes by or if he cannot keep up when sampling every
"nth" (every 3rd, 4th, etc.) case that passes by when sampling in a
line. Bias would be introduced because the sampler would tend to read
highly legible cases, but undersample cases with less legible dates.
Similarly, bias can be introduced in sampling from stacked cases if
the sampler is less than diligent about recording dates that are
difficult to read.
FAMRC researchers found that the line emerging from the case washer
was the best place to obtain a sample in most plants. As the cases
emerge from the washer, they are usually stacked five or six cases high.
A sampler can easily record dates from every case in a given stack.
If more time is required to examine a hard to read date, the case can
be removed from the line for closer inspection and another substituted.
Similarly, a whole stack maybe removed from the line for closer scrutiny.
Any number of stacks of cases can be ignored, but every case in a given
stack must be read to minimize bias.
The only type of case that should be ignored is a foreign case.
Foreign cases constituted an extremely small proportion of the cases
observed in cooperating firms' plants, but even so, foreign cases cannot
be analyzed because of the obvious lack of'acquisition data.
Analyzing the Sample: An Example
The case data shown in Table 2.is based on actual case acquisition
records and a sample of case manufacture dates observed at one of the
cooperating plants. The actual numbers of cases purchased have been
scaled (adjusted by a constant value) to maintain confidentiality, how
ever. Data in columns (1), (2), and (3) were obtained from plant records.
The plant was visited in early February and case manufacture dates
were recorded from a sample of 2,173 cases. Early February was chosen
as the time to draw a sample because the plant had acquired a shipment
of cases in early January; about a month had elapsed since the last
batch of cases had been acquired. A month was judged to be sufficient
to allow the newest cases to be thoroughly dispersed in the system.
Fortunately, case manufacture dates corresponded very closely with
the dates when cases were put into service. The numbers of cases observ
ed of each manufacture date were tabulated; the results are shown in
column (4) of Table 2. For example, in our sample of 2,173 cases, 145
were observed out of the shipment acquired with the date 181. Because
tabulation only requires counting, it is easily done by hand, but clerical
time can be saved by using a small computer if available.
After tabulating column (4), column (5) is calculated by dividing
the number of cases observed of a specific age by the number of cases
Table 2.An example of data and calculations required to determine case life.
(1)a (2) (3) (4) (5)b C6)c (7)d
Case
manufacture/ Months Number of Number of Sample Cases remaining
acquisition in cases cases ratio number
date system purchased in sample (4) (3) proportion (3) X (6)
181 1 787 145 .1842 1 787
980 5 3,637 532 .1463 .794 2,888
280 12 2,132 221 .1037 .563 1,200
979 17 4,929 401 .0814 .442 2,179
579 21 5,197 353 .0679 .369 1,918
1078 27 3,939 243 .0617 .335 1,320
578 33 3,443. 140 .0407 .221 761
1278 38 2,644 138 .0522 .283 748
Totals  26,708 2,173  11,801
a
Case manufacture dates are shown here; ideally they should correspond
they do not, column (21 should be adjusted to reflect as nearly as possible
put into service, rounded to the nearest month.
Column (5C is calculated by
of cases bought of the same age.
to acquisition
dates. If
the date cases were actually
dividing the number of cases observed of a specific age by the number
For example, 145787 = .1842, etc.
c
Column (6) is calculated by dividing every number in column (51 by the first number in column
(5L. For example, .1842L.1842 = 1, .1463t.1842= .794, etc. The resulting numbers represent the
proportions of cases of each batch that remain in the system. For percentages, multiply proportions
by 100.
d
Column (71 is calculated by multiplying the proportion of cases remaining in the system
by the number of cases originally purchased, i.e., columns (3) and (6).
bought or acquired of the same age. For example 145 1 787 = .1842,
532 7 3,637 = .1463, etc. The resulting numbers are termed the "sample
ratio" (Table 2). The sample ratios are then used to calculate column
(6), the proportion of cases of specific ages remaining in the system.
This technique has been used by another researcher toanalyze similar
problems (Hausman, 1979).
The values in column (6) are calculated by dividing every number in
column (5) by the first number in column (5) for example, .1842 .1842
= 1, .1463 .1842 = .794, etc. This calculation implicitly assumes
that none of the latest batch of cases is missing. The sample ratios
for all other batches of cases are expressed relative to the most recent
batch. The values calculated for column (6) represent the proportions of
cases remaining of each batch. For example, about 79 percent of the
cases are estimated to be in the system at the end of five months (those
acquired 980), but only 33.5 percent of those purchased 27 months
previously i.e., 1078 (Table 2). Obviously, to calculate the proportion
of cases missing, one would subtract the proportion remaining from 1.0.
The proportions are converted to percentages by multiplying by 100.
After the proportions of cases remaining in the system have been
estimated (olumn 6),estimates of actual case numbers of each age group
can be calculated by multiplying the values in column 6 by the numbers
of cases originally purchased (Column 3). These estimates are shown in
column (7). Summing the remaining cases of all ages yields an estimate
of total inventory. In Table 2, an estimated 11,801 cases remain of the
26,708 purchased in the last 38 months.
The values in column (6) can also be plotted to give a visual or
graphic representation of case losses over time (Figure 2). If an
Percent
100
90
80
70
60
50 Free hand curve
40
30
20 
10
 I '' S I j I I I '
1 5 12 17 21 27 33 38
Months
Figure 2. Percent of cases remaining in a distribution system over time (from Table 2, Column 6).
estimate of the percentage of cases remaining is desired for a point in
time between actual observations, a smooth, free hand curve can be drawn
through (among) the points, which attempts to minimize the deviations of
individual points (Figure 2). The approximate proportion of cases
remaining in the system can then be read off the graph for any time
period. If the values from column (6) are highly variable, it may be
difficult to draw an accurate, free hand curve through the resulting
plotted points. In such cases, the mathematical relationship between
the proportions of cases remaining (or lost) and time can be easily and
accurately estimated using a statistical procedure discussed in the
Appendix.
CONCLUSIONS
Data limitations precluded the determination of relative loss rates
for cases made of different materials or designs. However, the sampling
technique developed provides dairy processing plant managers with a
technique which can be used within their respective firms to estimate
case life for their own unique operations. It is important that managers
begin to have each new shipment of cases uniquely identified with legible
manufacture dates. The should also record the dates that cases are
actually put into service. This will allow case life to be monitored
using this procedure. An analysis of case life for different company
locations, case types, and management practices can help identify ways
to reduce case costs.
APPENDIX
APPENDIX
Case loss can be expressed as a mathematical function of the form
Y = ebt where:
Y = the ratio of remaining cases, i.e., column (.6) from Table 2.,
also shown in Appendix Table 1.
e = logarithmic base, 2.7182818, a constant.
t = time (in months) that various batches of cases have been in
the system, ice. column (2) Table 1, also shown in Appendix
Table 1. Other time units such as days or years could be
used if desired.
The objective of this procedure is to estimate the value of the
parameter "b" in the above equation by using a simplified form of re
gression analysis. The purpose of this mathematical approach is to be
able to calculate an "average" relationship between case loss and time,
resulting in a smooth, accurate curve such as the one shown in Appendix
Figure 1. This is especially useful when the calculated points (column
6) are highly variable. Points on the curve were calculated by allowing
"t" to take on values from 1 to 38 after "b" was estimated. To estimate
"b", the following steps are required:
1. Calculate columns (2) and (6) of Table 1, shown again in Appendix
Table 1.
2. Construct column (8) of Appendix Table 1 by taking the natural
log (base e) of column (6). Many inexpensive hand held cal
culators wtll perform this operation.
3. Add the squares of the entries in column (2):
12 + 52 + 122 + 172 + 212 + 272 + 333 + 382 = 4162
4. Add the products of the entries in column (2) times the entries
in column (8):
(1)0 + 5(.231) + 12(.574) + 17(.816) + 21(..997) + 27(.1.093)
+ 33(1.509) + 38(1.262) = 170.1
5. The parameter "b" equals the result in step 4 divided by the
result in step 3:
170.1 4162 = .04087
6. The equation is y = e'04087t
7. Cases remaining can now be estimated for any time period between
one and 38 months. For example, the estimated proportion of
cases remaining after 24 months in the system would be calculated
as follows:
Y = 2.7182818.04087(24)
Y = 2.7182818.98088
Y = .375 or 37.5 percent
8. The time required for a firm to lose a given proportion of cases
can also be calculated. The formula below will allow you to
solve for t given Y:
t = In Y
b
For example, to calculate the number of months it would take for
half the cases to leave the system, take the natural log of .50,
divid the resulting logarithm by .04087, the estimated "b" value
discussed above.
t = 0.693147 = 16.96 months
0.04087
Appendix Table l.An example of data and calculations required to
determine case life using a statistical procedure.
Columnb
(2) (6) (8)
Months in system Cases remaining, proportion Log of column (6)
1 1 0
5 .794 .231
12 .563 .574
17 .442 .816
21 .369 .997
27 .335 1.093
33 .221 1.509
38 .283 1.262
a
The statistical estimating procedure is regression analysis, used
to estimate the parameter b of the function Y = ebt where
Y = the ratio of remaining cases, column (6)
t = time (months) that various batches of cases have been in the
system, and
e = logarithmic base, 2.7182818.
b
Columns (2) and (6) are taken from Table 2 in the text. The
values in column (8) are natural logs (base e) of the values in column
(6).
Percent
100 
90
80
70 
Line drawn through points
60 calculated with mathematical
e function y = e04087t
50 Points from/
Column (6)
40 Appendix Table 1 ,
30
20
10
I I i i S S S S S S I I I  I l l l I  l I i i I 1 i I l '
1 5 12 17 21 27 33 38
Months
Appendix Figure 1. Percent of cases remaining in a distribution system over time.
REFERENCES
Mathis, Kary and Robert L. Degner, Dairy Delivery Case Losses in
Florida: Costs and Controls, Industry Report 771, Florida
Agricultural Market Research Center, Food and Resource Economics
Department, IFAS, University of Florida, June 1977.
Mathis, Kary and Robert L. Degner, Dairy Delivery Case Losses in
Florida 1979, Industry Report 796, Florida Agricultural Market
Research Center, Food and Resource Economics Department, IFAS,
University of Florida, November 1979.
Hausman, Jerry A. "Individual Discount Rates and the Purchase and
Utilization of Energy Using Durables", The Bell Journal of Economics,
Vol. 10, No. 1, Spring, 1979.
